CINXE.COM
(PDF) On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values
<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="Vu6yJUpmbk8dNW9j197nuunWn6xozwXmKO6xoycltxdSC_Av7NaJnIbcw64Ma5QJUpVw6WAqtcdc_LcXoKSoMQ" /> <meta name="citation_title" content="On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values" /> <meta name="citation_publication_date" content="2019/01/01" /> <meta name="citation_journal_title" content="Acta Mathematica Vietnamica" /> <meta name="citation_author" content="Nguyen Si Cuong (K17 HL)" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/94399355/On_the_Index_of_Reducibility_of_Parameter_Ideals_the_Stable_and_Limit_Values" /> <meta name="twitter:title" content="On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values" /> <meta name="twitter:description" content="Let (R, m) be a Noetherian local ring and M a finitely generated R-module of dimension d. A famous result of Northcott says that if M is Cohen-Macaulay, then the index of reducibility of parameter ideals on M is an invariant of the module. The aim of" /> <meta name="twitter:image" content="https://0.academia-photos.com/252120497/105438521/94638181/s200_nguyen_si_cuong._k17_hl_.png" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/94399355/On_the_Index_of_Reducibility_of_Parameter_Ideals_the_Stable_and_Limit_Values" /> <meta property="og:title" content="On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="Let (R, m) be a Noetherian local ring and M a finitely generated R-module of dimension d. A famous result of Northcott says that if M is Cohen-Macaulay, then the index of reducibility of parameter ideals on M is an invariant of the module. The aim of" /> <meta property="article:author" content="https://independent.academia.edu/NguyenSiCuongK17HL" /> <meta name="description" content="Let (R, m) be a Noetherian local ring and M a finitely generated R-module of dimension d. A famous result of Northcott says that if M is Cohen-Macaulay, then the index of reducibility of parameter ideals on M is an invariant of the module. The aim of" /> <title>(PDF) On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values</title> <link rel="canonical" href="https://www.academia.edu/94399355/On_the_Index_of_Reducibility_of_Parameter_Ideals_the_Stable_and_Limit_Values" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = 'b092bf3a3df71cf13feee7c143e83a57eb6b94fb'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1739833969000); window.Aedu.timeDifference = new Date().getTime() - 1739833969000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","author":[{"@context":"https://schema.org","@type":"Person","name":"Nguyen Si Cuong (K17 HL)","url":"https://independent.academia.edu/NguyenSiCuongK17HL"}],"contributor":[],"dateCreated":"2023-01-05","dateModified":"2024-11-22","datePublished":"2019-01-01","headline":"On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values","image":"https://attachments.academia-assets.com/96866615/thumbnails/1.jpg","inLanguage":"en","keywords":["Mathematics","Pure Mathematics","Limit (Mathematics)"],"publication":"Acta Mathematica Vietnamica","publisher":{"@context":"https://schema.org","@type":"Organization","name":"Springer Nature"},"sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":null}],"thumbnailUrl":"https://attachments.academia-assets.com/96866615/thumbnails/1.jpg","url":"https://www.academia.edu/94399355/On_the_Index_of_Reducibility_of_Parameter_Ideals_the_Stable_and_Limit_Values"}</script><style type="text/css">@media(max-width: 567px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: #cddaef;--button-primary-stroke-inverse-pressed: #0645b1;--button-secondary-stroke-inactive: #b1b1ba;--button-secondary-fill: #eef2f9;--button-secondary-text: #082f75;--button-secondary-fill-press: #cddaef;--button-secondary-fill-inactive: #ebebee;--button-secondary-stroke: #cddaef;--button-secondary-stroke-hover: #386ac1;--button-secondary-stroke-press: #0645b1;--button-secondary-text-inactive: #b1b1ba;--button-secondary-icon: #082f75;--button-secondary-fill-hover: #e6ecf7;--button-secondary-stroke-inverse: #ffffff;--button-secondary-fill-inverse: rgba(255, 255, 255, 0);--button-secondary-icon-inverse: #ffffff;--button-secondary-icon-hover: #082f75;--button-secondary-icon-press: #082f75;--button-secondary-text-inverse: #ffffff;--button-secondary-text-hover: #082f75;--button-secondary-text-press: #082f75;--button-secondary-fill-inverse-hover: #043059;--button-xs-stroke: #141413;--button-xs-stroke-hover: #0c3b8d;--button-xs-stroke-press: #082f75;--button-xs-stroke-inactive: #ebebee;--button-xs-text: #141413;--button-xs-text-hover: #0c3b8d;--button-xs-text-press: #082f75;--button-xs-text-inactive: #91919e;--button-xs-icon: #141413;--button-xs-icon-hover: #0c3b8d;--button-xs-icon-press: #082f75;--button-xs-icon-inactive: #91919e;--button-xs-fill: #ffffff;--button-xs-fill-hover: #f4f7fc;--button-xs-fill-press: #eef2f9;--buttons-button-text-inactive: #91919e;--buttons-button-focus: #0645b1;--buttons-button-icon-inactive: #91919e;--buttons-small-buttons-corner-radius: 16px;--buttons-small-buttons-l-r-padding: 20px;--buttons-small-buttons-height: 48px;--buttons-small-buttons-gap: 8px;--buttons-small-buttons-icon-only-width: 48px;--buttons-small-buttons-icon-size: 20px;--buttons-small-buttons-stroke-default: 1px;--buttons-small-buttons-stroke-thick: 2px;--buttons-large-buttons-l-r-padding: 32px;--buttons-large-buttons-height: 64px;--buttons-large-buttons-icon-only-width: 64px;--buttons-large-buttons-icon-size: 20px;--buttons-large-buttons-gap: 8px;--buttons-large-buttons-corner-radius: 16px;--buttons-large-buttons-stroke-default: 1px;--buttons-large-buttons-stroke-thick: 2px;--buttons-extra-small-buttons-l-r-padding: 8px;--buttons-extra-small-buttons-height: 32px;--buttons-extra-small-buttons-icon-size: 16px;--buttons-extra-small-buttons-gap: 4px;--buttons-extra-small-buttons-corner-radius: 8px;--buttons-stroke-default: 1px;--buttons-stroke-thick: 2px;--background-beige: #f9f7f4;--error-light: #fff2f2;--text-placeholder: #6d6d7d;--stroke-dark: #141413;--stroke-light: #dddde2;--stroke-medium: #535366;--accent-green: #ccffd4;--accent-turquoise: #ccf7ff;--accent-yellow: #f7ffcc;--accent-peach: #ffd4cc;--accent-violet: #f7ccff;--accent-purple: #f4f7fc;--text-primary: #141413;--secondary-brand: #141413;--text-hover: #0c3b8d;--text-white: #ffffff;--text-link: #0645b1;--text-press: #082f75;--success-light: #f0f8f1;--background-light-blue: #f4f7fc;--background-white: #ffffff;--premium-dark: #877440;--premium-light: #f9f6ed;--stroke-white: #ffffff;--inactive-content: #b1b1ba;--annotate-light: #a35dff;--annotate-dark: #824acc;--grid: #eef2f9;--inactive-stroke: #ebebee;--shadow: rgba(34, 34, 51, 0.25);--text-inactive: #6d6d7d;--text-error: #b60000;--stroke-error: #b60000;--background-error: #fff2f2;--background-black: #141413;--icon-default: #141413;--icon-blue: #0645b1;--background-grey: #dddde2;--icon-grey: #b1b1ba;--text-focus: #082f75;--brand-colors-neutral-black: #141413;--brand-colors-neutral-900: #535366;--brand-colors-neutral-800: #6d6d7d;--brand-colors-neutral-700: #91919e;--brand-colors-neutral-600: #b1b1ba;--brand-colors-neutral-500: #c8c8cf;--brand-colors-neutral-400: #dddde2;--brand-colors-neutral-300: #ebebee;--brand-colors-neutral-200: #f8f8fb;--brand-colors-neutral-100: #fafafa;--brand-colors-neutral-white: #ffffff;--brand-colors-blue-900: #043059;--brand-colors-blue-800: #082f75;--brand-colors-blue-700: #0c3b8d;--brand-colors-blue-600: #0645b1;--brand-colors-blue-500: #386ac1;--brand-colors-blue-400: #cddaef;--brand-colors-blue-300: #e6ecf7;--brand-colors-blue-200: #eef2f9;--brand-colors-blue-100: #f4f7fc;--brand-colors-gold-500: #877440;--brand-colors-gold-400: #e9e3d4;--brand-colors-gold-300: #f2efe8;--brand-colors-gold-200: #f9f6ed;--brand-colors-gold-100: #f9f7f4;--brand-colors-error-900: #920000;--brand-colors-error-500: #b60000;--brand-colors-success-900: #035c0f;--brand-colors-green: #ccffd4;--brand-colors-turquoise: #ccf7ff;--brand-colors-yellow: #f7ffcc;--brand-colors-peach: #ffd4cc;--brand-colors-violet: #f7ccff;--brand-colors-error-100: #fff2f2;--brand-colors-success-500: #05b01c;--brand-colors-success-100: #f0f8f1;--text-secondary: #535366;--icon-white: #ffffff;--background-beige-darker: #f2efe8;--icon-dark-grey: #535366;--type-font-family-sans-serif: DM Sans;--type-font-family-serif: Gupter;--type-font-family-mono: IBM Plex Mono;--type-weights-300: 300;--type-weights-400: 400;--type-weights-500: 500;--type-weights-700: 700;--type-sizes-12: 12px;--type-sizes-14: 14px;--type-sizes-16: 16px;--type-sizes-18: 18px;--type-sizes-20: 20px;--type-sizes-22: 22px;--type-sizes-24: 24px;--type-sizes-28: 28px;--type-sizes-30: 30px;--type-sizes-32: 32px;--type-sizes-40: 40px;--type-sizes-42: 42px;--type-sizes-48-2: 48px;--type-line-heights-16: 16px;--type-line-heights-20: 20px;--type-line-heights-23: 23px;--type-line-heights-24: 24px;--type-line-heights-25: 25px;--type-line-heights-26: 26px;--type-line-heights-29: 29px;--type-line-heights-30: 30px;--type-line-heights-32: 32px;--type-line-heights-34: 34px;--type-line-heights-35: 35px;--type-line-heights-36: 36px;--type-line-heights-38: 38px;--type-line-heights-40: 40px;--type-line-heights-46: 46px;--type-line-heights-48: 48px;--type-line-heights-52: 52px;--type-line-heights-58: 58px;--type-line-heights-68: 68px;--type-line-heights-74: 74px;--type-line-heights-82: 82px;--type-paragraph-spacings-0: 0px;--type-paragraph-spacings-4: 4px;--type-paragraph-spacings-8: 8px;--type-paragraph-spacings-16: 16px;--type-sans-serif-xl-font-weight: 400;--type-sans-serif-xl-size: 32px;--type-sans-serif-xl-line-height: 46px;--type-sans-serif-xl-paragraph-spacing: 16px;--type-sans-serif-lg-font-weight: 400;--type-sans-serif-lg-size: 30px;--type-sans-serif-lg-line-height: 36px;--type-sans-serif-lg-paragraph-spacing: 16px;--type-sans-serif-md-font-weight: 400;--type-sans-serif-md-line-height: 30px;--type-sans-serif-md-paragraph-spacing: 16px;--type-sans-serif-md-size: 24px;--type-sans-serif-xs-font-weight: 700;--type-sans-serif-xs-line-height: 24px;--type-sans-serif-xs-paragraph-spacing: 0px;--type-sans-serif-xs-size: 18px;--type-sans-serif-sm-font-weight: 400;--type-sans-serif-sm-line-height: 32px;--type-sans-serif-sm-paragraph-spacing: 16px;--type-sans-serif-sm-size: 20px;--type-body-xl-font-weight: 400;--type-body-xl-size: 24px;--type-body-xl-line-height: 36px;--type-body-xl-paragraph-spacing: 0px;--type-body-sm-font-weight: 400;--type-body-sm-size: 14px;--type-body-sm-line-height: 20px;--type-body-sm-paragraph-spacing: 8px;--type-body-xs-font-weight: 400;--type-body-xs-size: 12px;--type-body-xs-line-height: 16px;--type-body-xs-paragraph-spacing: 0px;--type-body-md-font-weight: 400;--type-body-md-size: 16px;--type-body-md-line-height: 20px;--type-body-md-paragraph-spacing: 4px;--type-body-lg-font-weight: 400;--type-body-lg-size: 20px;--type-body-lg-line-height: 26px;--type-body-lg-paragraph-spacing: 16px;--type-body-lg-medium-font-weight: 500;--type-body-lg-medium-size: 20px;--type-body-lg-medium-line-height: 32px;--type-body-lg-medium-paragraph-spacing: 16px;--type-body-md-medium-font-weight: 500;--type-body-md-medium-size: 16px;--type-body-md-medium-line-height: 20px;--type-body-md-medium-paragraph-spacing: 4px;--type-body-sm-bold-font-weight: 700;--type-body-sm-bold-size: 14px;--type-body-sm-bold-line-height: 20px;--type-body-sm-bold-paragraph-spacing: 8px;--type-body-sm-medium-font-weight: 500;--type-body-sm-medium-size: 14px;--type-body-sm-medium-line-height: 20px;--type-body-sm-medium-paragraph-spacing: 8px;--type-serif-md-font-weight: 400;--type-serif-md-size: 32px;--type-serif-md-paragraph-spacing: 0px;--type-serif-md-line-height: 40px;--type-serif-sm-font-weight: 400;--type-serif-sm-size: 24px;--type-serif-sm-paragraph-spacing: 0px;--type-serif-sm-line-height: 26px;--type-serif-lg-font-weight: 400;--type-serif-lg-size: 48px;--type-serif-lg-paragraph-spacing: 0px;--type-serif-lg-line-height: 52px;--type-serif-xs-font-weight: 400;--type-serif-xs-size: 18px;--type-serif-xs-line-height: 24px;--type-serif-xs-paragraph-spacing: 0px;--type-serif-xl-font-weight: 400;--type-serif-xl-size: 48px;--type-serif-xl-paragraph-spacing: 0px;--type-serif-xl-line-height: 58px;--type-mono-md-font-weight: 400;--type-mono-md-size: 22px;--type-mono-md-line-height: 24px;--type-mono-md-paragraph-spacing: 0px;--type-mono-lg-font-weight: 400;--type-mono-lg-size: 40px;--type-mono-lg-line-height: 40px;--type-mono-lg-paragraph-spacing: 0px;--type-mono-sm-font-weight: 400;--type-mono-sm-size: 14px;--type-mono-sm-line-height: 24px;--type-mono-sm-paragraph-spacing: 0px;--spacing-xs-4: 4px;--spacing-xs-8: 8px;--spacing-xs-16: 16px;--spacing-sm-24: 24px;--spacing-sm-32: 32px;--spacing-md-40: 40px;--spacing-md-48: 48px;--spacing-lg-64: 64px;--spacing-lg-80: 80px;--spacing-xlg-104: 104px;--spacing-xlg-152: 152px;--spacing-xs-12: 12px;--spacing-page-section: 80px;--spacing-card-list-spacing: 48px;--spacing-text-section-spacing: 64px;--spacing-md-xs-headings: 40px;--corner-radius-radius-lg: 16px;--corner-radius-radius-sm: 4px;--corner-radius-radius-md: 8px;--corner-radius-radius-round: 104px}}@media(min-width: 568px)and (max-width: 1279px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: #cddaef;--button-primary-stroke-inverse-pressed: #0645b1;--button-secondary-stroke-inactive: #b1b1ba;--button-secondary-fill: #eef2f9;--button-secondary-text: #082f75;--button-secondary-fill-press: #cddaef;--button-secondary-fill-inactive: #ebebee;--button-secondary-stroke: #cddaef;--button-secondary-stroke-hover: #386ac1;--button-secondary-stroke-press: #0645b1;--button-secondary-text-inactive: #b1b1ba;--button-secondary-icon: #082f75;--button-secondary-fill-hover: #e6ecf7;--button-secondary-stroke-inverse: #ffffff;--button-secondary-fill-inverse: rgba(255, 255, 255, 0);--button-secondary-icon-inverse: #ffffff;--button-secondary-icon-hover: #082f75;--button-secondary-icon-press: #082f75;--button-secondary-text-inverse: #ffffff;--button-secondary-text-hover: #082f75;--button-secondary-text-press: #082f75;--button-secondary-fill-inverse-hover: #043059;--button-xs-stroke: #141413;--button-xs-stroke-hover: #0c3b8d;--button-xs-stroke-press: #082f75;--button-xs-stroke-inactive: #ebebee;--button-xs-text: #141413;--button-xs-text-hover: #0c3b8d;--button-xs-text-press: #082f75;--button-xs-text-inactive: #91919e;--button-xs-icon: #141413;--button-xs-icon-hover: #0c3b8d;--button-xs-icon-press: #082f75;--button-xs-icon-inactive: #91919e;--button-xs-fill: #ffffff;--button-xs-fill-hover: #f4f7fc;--button-xs-fill-press: #eef2f9;--buttons-button-text-inactive: #91919e;--buttons-button-focus: #0645b1;--buttons-button-icon-inactive: #91919e;--buttons-small-buttons-corner-radius: 16px;--buttons-small-buttons-l-r-padding: 20px;--buttons-small-buttons-height: 48px;--buttons-small-buttons-gap: 8px;--buttons-small-buttons-icon-only-width: 48px;--buttons-small-buttons-icon-size: 20px;--buttons-small-buttons-stroke-default: 1px;--buttons-small-buttons-stroke-thick: 2px;--buttons-large-buttons-l-r-padding: 32px;--buttons-large-buttons-height: 64px;--buttons-large-buttons-icon-only-width: 64px;--buttons-large-buttons-icon-size: 20px;--buttons-large-buttons-gap: 8px;--buttons-large-buttons-corner-radius: 16px;--buttons-large-buttons-stroke-default: 1px;--buttons-large-buttons-stroke-thick: 2px;--buttons-extra-small-buttons-l-r-padding: 8px;--buttons-extra-small-buttons-height: 32px;--buttons-extra-small-buttons-icon-size: 16px;--buttons-extra-small-buttons-gap: 4px;--buttons-extra-small-buttons-corner-radius: 8px;--buttons-stroke-default: 1px;--buttons-stroke-thick: 2px;--background-beige: #f9f7f4;--error-light: #fff2f2;--text-placeholder: #6d6d7d;--stroke-dark: #141413;--stroke-light: #dddde2;--stroke-medium: #535366;--accent-green: #ccffd4;--accent-turquoise: #ccf7ff;--accent-yellow: #f7ffcc;--accent-peach: #ffd4cc;--accent-violet: #f7ccff;--accent-purple: #f4f7fc;--text-primary: #141413;--secondary-brand: #141413;--text-hover: #0c3b8d;--text-white: #ffffff;--text-link: #0645b1;--text-press: #082f75;--success-light: #f0f8f1;--background-light-blue: #f4f7fc;--background-white: #ffffff;--premium-dark: #877440;--premium-light: #f9f6ed;--stroke-white: #ffffff;--inactive-content: #b1b1ba;--annotate-light: #a35dff;--annotate-dark: #824acc;--grid: #eef2f9;--inactive-stroke: #ebebee;--shadow: rgba(34, 34, 51, 0.25);--text-inactive: #6d6d7d;--text-error: #b60000;--stroke-error: #b60000;--background-error: #fff2f2;--background-black: #141413;--icon-default: #141413;--icon-blue: #0645b1;--background-grey: #dddde2;--icon-grey: #b1b1ba;--text-focus: #082f75;--brand-colors-neutral-black: #141413;--brand-colors-neutral-900: #535366;--brand-colors-neutral-800: #6d6d7d;--brand-colors-neutral-700: #91919e;--brand-colors-neutral-600: #b1b1ba;--brand-colors-neutral-500: #c8c8cf;--brand-colors-neutral-400: #dddde2;--brand-colors-neutral-300: #ebebee;--brand-colors-neutral-200: #f8f8fb;--brand-colors-neutral-100: #fafafa;--brand-colors-neutral-white: #ffffff;--brand-colors-blue-900: #043059;--brand-colors-blue-800: #082f75;--brand-colors-blue-700: #0c3b8d;--brand-colors-blue-600: #0645b1;--brand-colors-blue-500: #386ac1;--brand-colors-blue-400: #cddaef;--brand-colors-blue-300: #e6ecf7;--brand-colors-blue-200: #eef2f9;--brand-colors-blue-100: #f4f7fc;--brand-colors-gold-500: #877440;--brand-colors-gold-400: #e9e3d4;--brand-colors-gold-300: #f2efe8;--brand-colors-gold-200: #f9f6ed;--brand-colors-gold-100: #f9f7f4;--brand-colors-error-900: #920000;--brand-colors-error-500: #b60000;--brand-colors-success-900: #035c0f;--brand-colors-green: #ccffd4;--brand-colors-turquoise: #ccf7ff;--brand-colors-yellow: #f7ffcc;--brand-colors-peach: #ffd4cc;--brand-colors-violet: #f7ccff;--brand-colors-error-100: #fff2f2;--brand-colors-success-500: #05b01c;--brand-colors-success-100: #f0f8f1;--text-secondary: #535366;--icon-white: #ffffff;--background-beige-darker: #f2efe8;--icon-dark-grey: #535366;--type-font-family-sans-serif: DM Sans;--type-font-family-serif: Gupter;--type-font-family-mono: IBM Plex Mono;--type-weights-300: 300;--type-weights-400: 400;--type-weights-500: 500;--type-weights-700: 700;--type-sizes-12: 12px;--type-sizes-14: 14px;--type-sizes-16: 16px;--type-sizes-18: 18px;--type-sizes-20: 20px;--type-sizes-22: 22px;--type-sizes-24: 24px;--type-sizes-28: 28px;--type-sizes-30: 30px;--type-sizes-32: 32px;--type-sizes-40: 40px;--type-sizes-42: 42px;--type-sizes-48-2: 48px;--type-line-heights-16: 16px;--type-line-heights-20: 20px;--type-line-heights-23: 23px;--type-line-heights-24: 24px;--type-line-heights-25: 25px;--type-line-heights-26: 26px;--type-line-heights-29: 29px;--type-line-heights-30: 30px;--type-line-heights-32: 32px;--type-line-heights-34: 34px;--type-line-heights-35: 35px;--type-line-heights-36: 36px;--type-line-heights-38: 38px;--type-line-heights-40: 40px;--type-line-heights-46: 46px;--type-line-heights-48: 48px;--type-line-heights-52: 52px;--type-line-heights-58: 58px;--type-line-heights-68: 68px;--type-line-heights-74: 74px;--type-line-heights-82: 82px;--type-paragraph-spacings-0: 0px;--type-paragraph-spacings-4: 4px;--type-paragraph-spacings-8: 8px;--type-paragraph-spacings-16: 16px;--type-sans-serif-xl-font-weight: 400;--type-sans-serif-xl-size: 42px;--type-sans-serif-xl-line-height: 46px;--type-sans-serif-xl-paragraph-spacing: 16px;--type-sans-serif-lg-font-weight: 400;--type-sans-serif-lg-size: 32px;--type-sans-serif-lg-line-height: 36px;--type-sans-serif-lg-paragraph-spacing: 16px;--type-sans-serif-md-font-weight: 400;--type-sans-serif-md-line-height: 34px;--type-sans-serif-md-paragraph-spacing: 16px;--type-sans-serif-md-size: 28px;--type-sans-serif-xs-font-weight: 700;--type-sans-serif-xs-line-height: 25px;--type-sans-serif-xs-paragraph-spacing: 0px;--type-sans-serif-xs-size: 20px;--type-sans-serif-sm-font-weight: 400;--type-sans-serif-sm-line-height: 30px;--type-sans-serif-sm-paragraph-spacing: 16px;--type-sans-serif-sm-size: 24px;--type-body-xl-font-weight: 400;--type-body-xl-size: 24px;--type-body-xl-line-height: 36px;--type-body-xl-paragraph-spacing: 0px;--type-body-sm-font-weight: 400;--type-body-sm-size: 14px;--type-body-sm-line-height: 20px;--type-body-sm-paragraph-spacing: 8px;--type-body-xs-font-weight: 400;--type-body-xs-size: 12px;--type-body-xs-line-height: 16px;--type-body-xs-paragraph-spacing: 0px;--type-body-md-font-weight: 400;--type-body-md-size: 16px;--type-body-md-line-height: 20px;--type-body-md-paragraph-spacing: 4px;--type-body-lg-font-weight: 400;--type-body-lg-size: 20px;--type-body-lg-line-height: 26px;--type-body-lg-paragraph-spacing: 16px;--type-body-lg-medium-font-weight: 500;--type-body-lg-medium-size: 20px;--type-body-lg-medium-line-height: 32px;--type-body-lg-medium-paragraph-spacing: 16px;--type-body-md-medium-font-weight: 500;--type-body-md-medium-size: 16px;--type-body-md-medium-line-height: 20px;--type-body-md-medium-paragraph-spacing: 4px;--type-body-sm-bold-font-weight: 700;--type-body-sm-bold-size: 14px;--type-body-sm-bold-line-height: 20px;--type-body-sm-bold-paragraph-spacing: 8px;--type-body-sm-medium-font-weight: 500;--type-body-sm-medium-size: 14px;--type-body-sm-medium-line-height: 20px;--type-body-sm-medium-paragraph-spacing: 8px;--type-serif-md-font-weight: 400;--type-serif-md-size: 40px;--type-serif-md-paragraph-spacing: 0px;--type-serif-md-line-height: 48px;--type-serif-sm-font-weight: 400;--type-serif-sm-size: 28px;--type-serif-sm-paragraph-spacing: 0px;--type-serif-sm-line-height: 32px;--type-serif-lg-font-weight: 400;--type-serif-lg-size: 58px;--type-serif-lg-paragraph-spacing: 0px;--type-serif-lg-line-height: 68px;--type-serif-xs-font-weight: 400;--type-serif-xs-size: 18px;--type-serif-xs-line-height: 24px;--type-serif-xs-paragraph-spacing: 0px;--type-serif-xl-font-weight: 400;--type-serif-xl-size: 74px;--type-serif-xl-paragraph-spacing: 0px;--type-serif-xl-line-height: 82px;--type-mono-md-font-weight: 400;--type-mono-md-size: 22px;--type-mono-md-line-height: 24px;--type-mono-md-paragraph-spacing: 0px;--type-mono-lg-font-weight: 400;--type-mono-lg-size: 40px;--type-mono-lg-line-height: 40px;--type-mono-lg-paragraph-spacing: 0px;--type-mono-sm-font-weight: 400;--type-mono-sm-size: 14px;--type-mono-sm-line-height: 24px;--type-mono-sm-paragraph-spacing: 0px;--spacing-xs-4: 4px;--spacing-xs-8: 8px;--spacing-xs-16: 16px;--spacing-sm-24: 24px;--spacing-sm-32: 32px;--spacing-md-40: 40px;--spacing-md-48: 48px;--spacing-lg-64: 64px;--spacing-lg-80: 80px;--spacing-xlg-104: 104px;--spacing-xlg-152: 152px;--spacing-xs-12: 12px;--spacing-page-section: 104px;--spacing-card-list-spacing: 48px;--spacing-text-section-spacing: 80px;--spacing-md-xs-headings: 40px;--corner-radius-radius-lg: 16px;--corner-radius-radius-sm: 4px;--corner-radius-radius-md: 8px;--corner-radius-radius-round: 104px}}@media(min-width: 1280px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: #cddaef;--button-primary-stroke-inverse-pressed: #0645b1;--button-secondary-stroke-inactive: #b1b1ba;--button-secondary-fill: #eef2f9;--button-secondary-text: #082f75;--button-secondary-fill-press: #cddaef;--button-secondary-fill-inactive: #ebebee;--button-secondary-stroke: #cddaef;--button-secondary-stroke-hover: #386ac1;--button-secondary-stroke-press: #0645b1;--button-secondary-text-inactive: #b1b1ba;--button-secondary-icon: #082f75;--button-secondary-fill-hover: #e6ecf7;--button-secondary-stroke-inverse: #ffffff;--button-secondary-fill-inverse: rgba(255, 255, 255, 0);--button-secondary-icon-inverse: #ffffff;--button-secondary-icon-hover: #082f75;--button-secondary-icon-press: #082f75;--button-secondary-text-inverse: #ffffff;--button-secondary-text-hover: #082f75;--button-secondary-text-press: #082f75;--button-secondary-fill-inverse-hover: #043059;--button-xs-stroke: #141413;--button-xs-stroke-hover: #0c3b8d;--button-xs-stroke-press: #082f75;--button-xs-stroke-inactive: #ebebee;--button-xs-text: #141413;--button-xs-text-hover: #0c3b8d;--button-xs-text-press: #082f75;--button-xs-text-inactive: #91919e;--button-xs-icon: #141413;--button-xs-icon-hover: #0c3b8d;--button-xs-icon-press: #082f75;--button-xs-icon-inactive: #91919e;--button-xs-fill: #ffffff;--button-xs-fill-hover: #f4f7fc;--button-xs-fill-press: #eef2f9;--buttons-button-text-inactive: #91919e;--buttons-button-focus: #0645b1;--buttons-button-icon-inactive: #91919e;--buttons-small-buttons-corner-radius: 16px;--buttons-small-buttons-l-r-padding: 20px;--buttons-small-buttons-height: 48px;--buttons-small-buttons-gap: 8px;--buttons-small-buttons-icon-only-width: 48px;--buttons-small-buttons-icon-size: 20px;--buttons-small-buttons-stroke-default: 1px;--buttons-small-buttons-stroke-thick: 2px;--buttons-large-buttons-l-r-padding: 32px;--buttons-large-buttons-height: 64px;--buttons-large-buttons-icon-only-width: 64px;--buttons-large-buttons-icon-size: 20px;--buttons-large-buttons-gap: 8px;--buttons-large-buttons-corner-radius: 16px;--buttons-large-buttons-stroke-default: 1px;--buttons-large-buttons-stroke-thick: 2px;--buttons-extra-small-buttons-l-r-padding: 8px;--buttons-extra-small-buttons-height: 32px;--buttons-extra-small-buttons-icon-size: 16px;--buttons-extra-small-buttons-gap: 4px;--buttons-extra-small-buttons-corner-radius: 8px;--buttons-stroke-default: 1px;--buttons-stroke-thick: 2px;--background-beige: #f9f7f4;--error-light: #fff2f2;--text-placeholder: #6d6d7d;--stroke-dark: #141413;--stroke-light: #dddde2;--stroke-medium: #535366;--accent-green: #ccffd4;--accent-turquoise: #ccf7ff;--accent-yellow: #f7ffcc;--accent-peach: #ffd4cc;--accent-violet: #f7ccff;--accent-purple: #f4f7fc;--text-primary: #141413;--secondary-brand: #141413;--text-hover: #0c3b8d;--text-white: #ffffff;--text-link: #0645b1;--text-press: #082f75;--success-light: #f0f8f1;--background-light-blue: #f4f7fc;--background-white: #ffffff;--premium-dark: #877440;--premium-light: #f9f6ed;--stroke-white: #ffffff;--inactive-content: #b1b1ba;--annotate-light: #a35dff;--annotate-dark: #824acc;--grid: #eef2f9;--inactive-stroke: #ebebee;--shadow: rgba(34, 34, 51, 0.25);--text-inactive: #6d6d7d;--text-error: #b60000;--stroke-error: #b60000;--background-error: #fff2f2;--background-black: #141413;--icon-default: #141413;--icon-blue: #0645b1;--background-grey: #dddde2;--icon-grey: #b1b1ba;--text-focus: #082f75;--brand-colors-neutral-black: #141413;--brand-colors-neutral-900: #535366;--brand-colors-neutral-800: #6d6d7d;--brand-colors-neutral-700: #91919e;--brand-colors-neutral-600: #b1b1ba;--brand-colors-neutral-500: #c8c8cf;--brand-colors-neutral-400: #dddde2;--brand-colors-neutral-300: #ebebee;--brand-colors-neutral-200: #f8f8fb;--brand-colors-neutral-100: #fafafa;--brand-colors-neutral-white: #ffffff;--brand-colors-blue-900: #043059;--brand-colors-blue-800: #082f75;--brand-colors-blue-700: #0c3b8d;--brand-colors-blue-600: #0645b1;--brand-colors-blue-500: #386ac1;--brand-colors-blue-400: #cddaef;--brand-colors-blue-300: #e6ecf7;--brand-colors-blue-200: #eef2f9;--brand-colors-blue-100: #f4f7fc;--brand-colors-gold-500: #877440;--brand-colors-gold-400: #e9e3d4;--brand-colors-gold-300: #f2efe8;--brand-colors-gold-200: #f9f6ed;--brand-colors-gold-100: #f9f7f4;--brand-colors-error-900: #920000;--brand-colors-error-500: #b60000;--brand-colors-success-900: #035c0f;--brand-colors-green: #ccffd4;--brand-colors-turquoise: #ccf7ff;--brand-colors-yellow: #f7ffcc;--brand-colors-peach: #ffd4cc;--brand-colors-violet: #f7ccff;--brand-colors-error-100: #fff2f2;--brand-colors-success-500: #05b01c;--brand-colors-success-100: #f0f8f1;--text-secondary: #535366;--icon-white: #ffffff;--background-beige-darker: #f2efe8;--icon-dark-grey: #535366;--type-font-family-sans-serif: DM Sans;--type-font-family-serif: Gupter;--type-font-family-mono: IBM Plex Mono;--type-weights-300: 300;--type-weights-400: 400;--type-weights-500: 500;--type-weights-700: 700;--type-sizes-12: 12px;--type-sizes-14: 14px;--type-sizes-16: 16px;--type-sizes-18: 18px;--type-sizes-20: 20px;--type-sizes-22: 22px;--type-sizes-24: 24px;--type-sizes-28: 28px;--type-sizes-30: 30px;--type-sizes-32: 32px;--type-sizes-40: 40px;--type-sizes-42: 42px;--type-sizes-48-2: 48px;--type-line-heights-16: 16px;--type-line-heights-20: 20px;--type-line-heights-23: 23px;--type-line-heights-24: 24px;--type-line-heights-25: 25px;--type-line-heights-26: 26px;--type-line-heights-29: 29px;--type-line-heights-30: 30px;--type-line-heights-32: 32px;--type-line-heights-34: 34px;--type-line-heights-35: 35px;--type-line-heights-36: 36px;--type-line-heights-38: 38px;--type-line-heights-40: 40px;--type-line-heights-46: 46px;--type-line-heights-48: 48px;--type-line-heights-52: 52px;--type-line-heights-58: 58px;--type-line-heights-68: 68px;--type-line-heights-74: 74px;--type-line-heights-82: 82px;--type-paragraph-spacings-0: 0px;--type-paragraph-spacings-4: 4px;--type-paragraph-spacings-8: 8px;--type-paragraph-spacings-16: 16px;--type-sans-serif-xl-font-weight: 400;--type-sans-serif-xl-size: 42px;--type-sans-serif-xl-line-height: 46px;--type-sans-serif-xl-paragraph-spacing: 16px;--type-sans-serif-lg-font-weight: 400;--type-sans-serif-lg-size: 32px;--type-sans-serif-lg-line-height: 38px;--type-sans-serif-lg-paragraph-spacing: 16px;--type-sans-serif-md-font-weight: 400;--type-sans-serif-md-line-height: 34px;--type-sans-serif-md-paragraph-spacing: 16px;--type-sans-serif-md-size: 28px;--type-sans-serif-xs-font-weight: 700;--type-sans-serif-xs-line-height: 25px;--type-sans-serif-xs-paragraph-spacing: 0px;--type-sans-serif-xs-size: 20px;--type-sans-serif-sm-font-weight: 400;--type-sans-serif-sm-line-height: 30px;--type-sans-serif-sm-paragraph-spacing: 16px;--type-sans-serif-sm-size: 24px;--type-body-xl-font-weight: 400;--type-body-xl-size: 24px;--type-body-xl-line-height: 36px;--type-body-xl-paragraph-spacing: 0px;--type-body-sm-font-weight: 400;--type-body-sm-size: 14px;--type-body-sm-line-height: 20px;--type-body-sm-paragraph-spacing: 8px;--type-body-xs-font-weight: 400;--type-body-xs-size: 12px;--type-body-xs-line-height: 16px;--type-body-xs-paragraph-spacing: 0px;--type-body-md-font-weight: 400;--type-body-md-size: 16px;--type-body-md-line-height: 20px;--type-body-md-paragraph-spacing: 4px;--type-body-lg-font-weight: 400;--type-body-lg-size: 20px;--type-body-lg-line-height: 26px;--type-body-lg-paragraph-spacing: 16px;--type-body-lg-medium-font-weight: 500;--type-body-lg-medium-size: 20px;--type-body-lg-medium-line-height: 32px;--type-body-lg-medium-paragraph-spacing: 16px;--type-body-md-medium-font-weight: 500;--type-body-md-medium-size: 16px;--type-body-md-medium-line-height: 20px;--type-body-md-medium-paragraph-spacing: 4px;--type-body-sm-bold-font-weight: 700;--type-body-sm-bold-size: 14px;--type-body-sm-bold-line-height: 20px;--type-body-sm-bold-paragraph-spacing: 8px;--type-body-sm-medium-font-weight: 500;--type-body-sm-medium-size: 14px;--type-body-sm-medium-line-height: 20px;--type-body-sm-medium-paragraph-spacing: 8px;--type-serif-md-font-weight: 400;--type-serif-md-size: 40px;--type-serif-md-paragraph-spacing: 0px;--type-serif-md-line-height: 48px;--type-serif-sm-font-weight: 400;--type-serif-sm-size: 28px;--type-serif-sm-paragraph-spacing: 0px;--type-serif-sm-line-height: 32px;--type-serif-lg-font-weight: 400;--type-serif-lg-size: 58px;--type-serif-lg-paragraph-spacing: 0px;--type-serif-lg-line-height: 68px;--type-serif-xs-font-weight: 400;--type-serif-xs-size: 18px;--type-serif-xs-line-height: 24px;--type-serif-xs-paragraph-spacing: 0px;--type-serif-xl-font-weight: 400;--type-serif-xl-size: 74px;--type-serif-xl-paragraph-spacing: 0px;--type-serif-xl-line-height: 82px;--type-mono-md-font-weight: 400;--type-mono-md-size: 22px;--type-mono-md-line-height: 24px;--type-mono-md-paragraph-spacing: 0px;--type-mono-lg-font-weight: 400;--type-mono-lg-size: 40px;--type-mono-lg-line-height: 40px;--type-mono-lg-paragraph-spacing: 0px;--type-mono-sm-font-weight: 400;--type-mono-sm-size: 14px;--type-mono-sm-line-height: 24px;--type-mono-sm-paragraph-spacing: 0px;--spacing-xs-4: 4px;--spacing-xs-8: 8px;--spacing-xs-16: 16px;--spacing-sm-24: 24px;--spacing-sm-32: 32px;--spacing-md-40: 40px;--spacing-md-48: 48px;--spacing-lg-64: 64px;--spacing-lg-80: 80px;--spacing-xlg-104: 104px;--spacing-xlg-152: 152px;--spacing-xs-12: 12px;--spacing-page-section: 152px;--spacing-card-list-spacing: 48px;--spacing-text-section-spacing: 80px;--spacing-md-xs-headings: 40px;--corner-radius-radius-lg: 16px;--corner-radius-radius-sm: 4px;--corner-radius-radius-md: 8px;--corner-radius-radius-round: 104px}}</style><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/single_work_page/loswp-f29774c14b4e629cbb4375c919ad1c2b2891ac825d0a410ea6339ae17e481a55.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/body-170d1319f0e354621e81ca17054bb147da2856ec0702fe440a99af314a6338c5.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/button-bfbac2a470372e2f3a6661a65fa7ff0a0fbf7aa32534d9a831d683d2a6f9e01b.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/heading-95367dc03b794f6737f30123738a886cf53b7a65cdef98a922a98591d60063e3.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/text_button-d1941ab08e91e29ee143084c4749da4aaffa350a2ac6eec2306b1d7a352d911a.css" /><link crossorigin="" href="https://fonts.gstatic.com/" rel="preconnect" /><link href="https://fonts.googleapis.com/css2?family=DM+Sans:ital,opsz,wght@0,9..40,100..1000;1,9..40,100..1000&family=Gupter:wght@400;500;700&family=IBM+Plex+Mono:wght@300;400&family=Material+Symbols+Outlined:opsz,wght,FILL,GRAD@20,400,0,0&display=swap" rel="stylesheet" /> </head> <body> <div id='react-modal'></div> <div class="js-upgrade-ie-banner" style="display: none; text-align: center; padding: 8px 0; background-color: #ebe480;"><p style="color: #000; font-size: 12px; margin: 0 0 4px;">Academia.edu no longer supports Internet Explorer.</p><p style="color: #000; font-size: 12px; margin: 0;">To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to <a href="https://www.academia.edu/upgrade-browser">upgrade your browser</a>.</p></div><script>// Show this banner for all versions of IE if (!!window.MSInputMethodContext || /(MSIE)/.test(navigator.userAgent)) { document.querySelector('.js-upgrade-ie-banner').style.display = 'block'; }</script> <div class="bootstrap login"><div class="modal fade login-modal" id="login-modal"><div class="login-modal-dialog modal-dialog"><div class="modal-content"><div class="modal-header"><button class="close close" data-dismiss="modal" type="button"><span aria-hidden="true">×</span><span class="sr-only">Close</span></button><h4 class="modal-title text-center"><strong>Log In</strong></h4></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><button class="btn btn-fb btn-lg btn-block btn-v-center-content" id="login-facebook-oauth-button"><svg style="float: left; width: 19px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="facebook-square" class="svg-inline--fa fa-facebook-square fa-w-14" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><path fill="currentColor" d="M400 32H48A48 48 0 0 0 0 80v352a48 48 0 0 0 48 48h137.25V327.69h-63V256h63v-54.64c0-62.15 37-96.48 93.67-96.48 27.14 0 55.52 4.84 55.52 4.84v61h-31.27c-30.81 0-40.42 19.12-40.42 38.73V256h68.78l-11 71.69h-57.78V480H400a48 48 0 0 0 48-48V80a48 48 0 0 0-48-48z"></path></svg><small><strong>Log in</strong> with <strong>Facebook</strong></small></button><br /><button class="btn btn-google btn-lg btn-block btn-v-center-content" id="login-google-oauth-button"><svg style="float: left; width: 22px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="google-plus" class="svg-inline--fa fa-google-plus fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256,8C119.1,8,8,119.1,8,256S119.1,504,256,504,504,392.9,504,256,392.9,8,256,8ZM185.3,380a124,124,0,0,1,0-248c31.3,0,60.1,11,83,32.3l-33.6,32.6c-13.2-12.9-31.3-19.1-49.4-19.1-42.9,0-77.2,35.5-77.2,78.1S142.3,334,185.3,334c32.6,0,64.9-19.1,70.1-53.3H185.3V238.1H302.2a109.2,109.2,0,0,1,1.9,20.7c0,70.8-47.5,121.2-118.8,121.2ZM415.5,273.8v35.5H380V273.8H344.5V238.3H380V202.8h35.5v35.5h35.2v35.5Z"></path></svg><small><strong>Log in</strong> with <strong>Google</strong></small></button><br /><style type="text/css">.sign-in-with-apple-button { width: 100%; height: 52px; border-radius: 3px; border: 1px solid black; cursor: pointer; } .sign-in-with-apple-button > div { margin: 0 auto; / This centers the Apple-rendered button horizontally }</style><script src="https://appleid.cdn-apple.com/appleauth/static/jsapi/appleid/1/en_US/appleid.auth.js" type="text/javascript"></script><div class="sign-in-with-apple-button" data-border="false" data-color="white" id="appleid-signin"><span ="Sign Up with Apple" class="u-fs11"></span></div><script>AppleID.auth.init({ clientId: 'edu.academia.applesignon', scope: 'name email', redirectURI: 'https://www.academia.edu/sessions', state: "9a019e39969920f28f10b9b76eed993d7cbdd1e3f19760c411da3ff5bb25172b", });</script><script>// Hacky way of checking if on fast loswp if (window.loswp == null) { (function() { const Google = window?.Aedu?.Auth?.OauthButton?.Login?.Google; const Facebook = window?.Aedu?.Auth?.OauthButton?.Login?.Facebook; if (Google) { new Google({ el: '#login-google-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } if (Facebook) { new Facebook({ el: '#login-facebook-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } })(); }</script></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><div class="hr-heading login-hr-heading"><span class="hr-heading-text">or</span></div></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><form class="js-login-form" action="https://www.academia.edu/sessions" accept-charset="UTF-8" method="post"><input type="hidden" name="authenticity_token" value="YdL9iq3HFq8g3RrG5HVN4j7-r8e4x3-ExbMbrLvt489lN7-AC3fxfLs0tgs_wD5Rhb1AgrAiz6WxoR0YPGz86Q" autocomplete="off" /><div class="form-group"><label class="control-label" for="login-modal-email-input" style="font-size: 14px;">Email</label><input class="form-control" id="login-modal-email-input" name="login" type="email" /></div><div class="form-group"><label class="control-label" for="login-modal-password-input" style="font-size: 14px;">Password</label><input class="form-control" id="login-modal-password-input" name="password" type="password" /></div><input type="hidden" name="post_login_redirect_url" id="post_login_redirect_url" value="https://www.academia.edu/94399355/On_the_Index_of_Reducibility_of_Parameter_Ideals_the_Stable_and_Limit_Values" autocomplete="off" /><div class="checkbox"><label><input type="checkbox" name="remember_me" id="remember_me" value="1" checked="checked" /><small style="font-size: 12px; margin-top: 2px; display: inline-block;">Remember me on this computer</small></label></div><br><input type="submit" name="commit" value="Log In" class="btn btn-primary btn-block btn-lg js-login-submit" data-disable-with="Log In" /></br></form><script>typeof window?.Aedu?.recaptchaManagedForm === 'function' && window.Aedu.recaptchaManagedForm( document.querySelector('.js-login-form'), document.querySelector('.js-login-submit') );</script><small style="font-size: 12px;"><br />or <a data-target="#login-modal-reset-password-container" data-toggle="collapse" href="javascript:void(0)">reset password</a></small><div class="collapse" id="login-modal-reset-password-container"><br /><div class="well margin-0x"><form class="js-password-reset-form" action="https://www.academia.edu/reset_password" accept-charset="UTF-8" method="post"><input type="hidden" name="authenticity_token" value="8OfJTt-FqQhWSJYQZCdKdOQVkfqJgfAXfRvHAjA-iYv0AotEeTVO282hOt2_kjnHX1Z-v4FkQDYJCcG2t7-WrQ" autocomplete="off" /><p>Enter the email address you signed up with and we'll email you a reset link.</p><div class="form-group"><input class="form-control" name="email" type="email" /></div><input class="btn btn-primary btn-block g-recaptcha js-password-reset-submit" data-sitekey="6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj" type="submit" value="Email me a link" /></form></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/collapse-45805421cf446ca5adf7aaa1935b08a3a8d1d9a6cc5d91a62a2a3a00b20b3e6a.js"], function() { // from javascript_helper.rb $("#login-modal-reset-password-container").on("shown.bs.collapse", function() { $(this).find("input[type=email]").focus(); }); }); </script> </div></div></div><div class="modal-footer"><div class="text-center"><small style="font-size: 12px;">Need an account? <a rel="nofollow" href="https://www.academia.edu/signup">Click here to sign up</a></small></div></div></div></div></div></div><script>// If we are on subdomain or non-bootstrapped page, redirect to login page instead of showing modal (function(){ if (typeof $ === 'undefined') return; var host = window.location.hostname; if ((host === $domain || host === "www."+$domain) && (typeof $().modal === 'function')) { $("#nav_log_in").click(function(e) { // Don't follow the link and open the modal e.preventDefault(); $("#login-modal").on('shown.bs.modal', function() { $(this).find("#login-modal-email-input").focus() }).modal('show'); }); } })()</script> <div id="fb-root"></div><script>window.fbAsyncInit = function() { FB.init({ appId: "2369844204", version: "v8.0", status: true, cookie: true, xfbml: true }); // Additional initialization code. if (window.InitFacebook) { // facebook.ts already loaded, set it up. window.InitFacebook(); } else { // Set a flag for facebook.ts to find when it loads. window.academiaAuthReadyFacebook = true; } };</script> <div id="google-root"></div><script>window.loadGoogle = function() { if (window.InitGoogle) { // google.ts already loaded, set it up. window.InitGoogle("331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"); } else { // Set a flag for google.ts to use when it loads. window.GoogleClientID = "331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"; } };</script> <div class="header--container" id="main-header-container"><div class="header--inner-container header--inner-container-ds2"><div class="header-ds2--left-wrapper"><div class="header-ds2--left-wrapper-inner"><a data-main-header-link-target="logo_home" href="https://www.academia.edu/"><img class="hide-on-desktop-redesign" style="height: 24px; width: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015-A.svg" width="24" height="24" /><img width="145.2" height="18" class="hide-on-mobile-redesign" style="height: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015.svg" /></a><div class="header--search-container header--search-container-ds2"><form class="js-SiteSearch-form select2-no-default-pills" action="https://www.academia.edu/search" accept-charset="UTF-8" method="get"><svg style="width: 14px; height: 14px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="search" class="header--search-icon svg-inline--fa fa-search fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M505 442.7L405.3 343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3 44-79.7 44-128C416 93.1 322.9 0 208 0S0 93.1 0 208s93.1 208 208 208c48.3 0 92.7-16.4 128-44v16.3c0 6.4 2.5 12.5 7 17l99.7 99.7c9.4 9.4 24.6 9.4 33.9 0l28.3-28.3c9.4-9.4 9.4-24.6.1-34zM208 336c-70.7 0-128-57.2-128-128 0-70.7 57.2-128 128-128 70.7 0 128 57.2 128 128 0 70.7-57.2 128-128 128z"></path></svg><input class="header--search-input header--search-input-ds2 js-SiteSearch-form-input" data-main-header-click-target="search_input" name="q" placeholder="Search" type="text" /></form></div></div></div><nav class="header--nav-buttons header--nav-buttons-ds2 js-main-nav"><button class="ds2-5-button ds2-5-button--secondary js-header-login-url header-button-ds2 header-login-ds2 hide-on-mobile-redesign react-login-modal-opener" data-signup-modal="{"location":"login-button--header"}" rel="nofollow">Log In</button><button class="ds2-5-button ds2-5-button--secondary header-button-ds2 hide-on-mobile-redesign react-login-modal-opener" data-signup-modal="{"location":"signup-button--header"}" rel="nofollow">Sign Up</button><button class="header--hamburger-button header--hamburger-button-ds2 hide-on-desktop-redesign js-header-hamburger-button"><div class="icon-bar"></div><div class="icon-bar" style="margin-top: 4px;"></div><div class="icon-bar" style="margin-top: 4px;"></div></button></nav></div><ul class="header--dropdown-container js-header-dropdown"><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/login" rel="nofollow">Log In</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/signup" rel="nofollow">Sign Up</a></li><li class="header--dropdown-row js-header-dropdown-expand-button"><button class="header--dropdown-button">more<svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="caret-down" class="header--dropdown-button-icon svg-inline--fa fa-caret-down fa-w-10" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path fill="currentColor" d="M31.3 192h257.3c17.8 0 26.7 21.5 14.1 34.1L174.1 354.8c-7.8 7.8-20.5 7.8-28.3 0L17.2 226.1C4.6 213.5 13.5 192 31.3 192z"></path></svg></button></li><li><ul class="header--expanded-dropdown-container"><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/about">About</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/press">Press</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/documents">Papers</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/terms">Terms</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/privacy">Privacy</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/copyright">Copyright</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/hiring"><svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="briefcase" class="header--dropdown-row-icon svg-inline--fa fa-briefcase fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M320 336c0 8.84-7.16 16-16 16h-96c-8.84 0-16-7.16-16-16v-48H0v144c0 25.6 22.4 48 48 48h416c25.6 0 48-22.4 48-48V288H320v48zm144-208h-80V80c0-25.6-22.4-48-48-48H176c-25.6 0-48 22.4-48 48v48H48c-25.6 0-48 22.4-48 48v80h512v-80c0-25.6-22.4-48-48-48zm-144 0H192V96h128v32z"></path></svg>We're Hiring!</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://support.academia.edu/hc/en-us"><svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="question-circle" class="header--dropdown-row-icon svg-inline--fa fa-question-circle fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M504 256c0 136.997-111.043 248-248 248S8 392.997 8 256C8 119.083 119.043 8 256 8s248 111.083 248 248zM262.655 90c-54.497 0-89.255 22.957-116.549 63.758-3.536 5.286-2.353 12.415 2.715 16.258l34.699 26.31c5.205 3.947 12.621 3.008 16.665-2.122 17.864-22.658 30.113-35.797 57.303-35.797 20.429 0 45.698 13.148 45.698 32.958 0 14.976-12.363 22.667-32.534 33.976C247.128 238.528 216 254.941 216 296v4c0 6.627 5.373 12 12 12h56c6.627 0 12-5.373 12-12v-1.333c0-28.462 83.186-29.647 83.186-106.667 0-58.002-60.165-102-116.531-102zM256 338c-25.365 0-46 20.635-46 46 0 25.364 20.635 46 46 46s46-20.636 46-46c0-25.365-20.635-46-46-46z"></path></svg>Help Center</a></li><li class="header--dropdown-row js-header-dropdown-collapse-button"><button class="header--dropdown-button">less<svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="caret-up" class="header--dropdown-button-icon svg-inline--fa fa-caret-up fa-w-10" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path fill="currentColor" d="M288.662 352H31.338c-17.818 0-26.741-21.543-14.142-34.142l128.662-128.662c7.81-7.81 20.474-7.81 28.284 0l128.662 128.662c12.6 12.599 3.676 34.142-14.142 34.142z"></path></svg></button></li></ul></li></ul></div> <script src="//a.academia-assets.com/assets/webpack_bundles/fast_loswp-bundle-e5ca05062a7092a8f6f5af11f70589210af26f6a0030f102b0b21b22451b9d41.js" defer="defer"></script><script>window.loswp = {}; window.loswp.author = 252120497; window.loswp.bulkDownloadFilterCounts = {}; window.loswp.hasDownloadableAttachment = true; window.loswp.hasViewableAttachments = true; // TODO: just use routes for this window.loswp.loginUrl = "https://www.academia.edu/login?post_login_redirect_url=https%3A%2F%2Fwww.academia.edu%2F94399355%2FOn_the_Index_of_Reducibility_of_Parameter_Ideals_the_Stable_and_Limit_Values%3Fauto%3Ddownload"; window.loswp.translateUrl = "https://www.academia.edu/login?post_login_redirect_url=https%3A%2F%2Fwww.academia.edu%2F94399355%2FOn_the_Index_of_Reducibility_of_Parameter_Ideals_the_Stable_and_Limit_Values%3Fshow_translation%3Dtrue"; window.loswp.previewableAttachments = [{"id":96866615,"identifier":"Attachment_96866615","shouldShowBulkDownload":false}]; window.loswp.shouldDetectTimezone = true; window.loswp.shouldShowBulkDownload = true; window.loswp.showSignupCaptcha = false window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":94399355,"created_at":"2023-01-05T08:00:03.554-08:00","from_world_paper_id":224604454,"updated_at":"2024-11-22T21:16:24.157-08:00","_data":{"publisher":"Springer Nature","grobid_abstract":"Let (R, m) be a Noetherian local ring and M a finitely generated R-module of dimension d. A famous result of Northcott says that if M is Cohen-Macaulay, then the index of reducibility of parameter ideals on M is an invariant of the module. The aim of this paper is to extend Northcott's theorem for any finitely generated R-module. We call this invariant the stable value of the indices of reducibility of parameter ideals of M. We also introduce the limit value of the indices of reducibility of parameter ideals of M. Recently, the above results were extended for the classes of sequentially (generalized) Cohen-Macaulay modules and for parameter ideals generated by good systems of parameters in [17, 20].","publication_date":"2019,,","publication_name":"Acta Mathematica Vietnamica","grobid_abstract_attachment_id":"96866615"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values","broadcastable":false,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [252120497]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "full_page_mobile_sutd_modal"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":96866615,"attachmentType":"pdf"}"><img alt="First page of “On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/96866615/mini_magick20230105-1-1p13bwp.png?1672938097" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">On the Index of Reducibility of Parameter Ideals: the Stable and Limit Values</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="252120497" href="https://independent.academia.edu/NguyenSiCuongK17HL"><img alt="Profile image of Nguyen Si Cuong (K17 HL)" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/252120497/105438521/94638181/s65_nguyen_si_cuong._k17_hl_.png" />Nguyen Si Cuong (K17 HL)</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2019, Acta Mathematica Vietnamica</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">11 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 94399355; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=94399355"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">Let (R, m) be a Noetherian local ring and M a finitely generated R-module of dimension d. A famous result of Northcott says that if M is Cohen-Macaulay, then the index of reducibility of parameter ideals on M is an invariant of the module. The aim of this paper is to extend Northcott's theorem for any finitely generated R-module. We call this invariant the stable value of the indices of reducibility of parameter ideals of M. We also introduce the limit value of the indices of reducibility of parameter ideals of M. Recently, the above results were extended for the classes of sequentially (generalized) Cohen-Macaulay modules and for parameter ideals generated by good systems of parameters in [17, 20].</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":96866615,"attachmentType":"pdf","workUrl":"https://www.academia.edu/94399355/On_the_Index_of_Reducibility_of_Parameter_Ideals_the_Stable_and_Limit_Values"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":96866615,"attachmentType":"pdf","workUrl":"https://www.academia.edu/94399355/On_the_Index_of_Reducibility_of_Parameter_Ideals_the_Stable_and_Limit_Values"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. If the signup banner trigger (a 242px-high* invisible div underneath the 'See Full PDF' / 'Download PDF' buttons) // is already fully scrolled above the viewport, show the banner by default // 2. If the signup banner trigger is fully visible, show the banner // 3. If the signup banner trigger has even a few pixels scrolled below the viewport, hide the banner // // * 242px is the empirically determined height of the signup banner. It's better to be a bit taller than // necessary than too short, so it's fine that the mobile (small breakpoint) banner is shorter. // First check session storage for the signup banner's visibility state const signupBannerHidden = sessionStorage.getItem('ds-signup-banner-hidden'); if (signupBannerHidden === 'true') { return; } const signupBanner = document.querySelector('.ds-signup-banner'); const signupBannerTrigger = document.querySelector('.ds-signup-banner-trigger'); if (!signupBannerTrigger) { window.Sentry.captureMessage("Signup banner trigger not found"); return; } let footerShown = false; window.addEventListener('load', () => { const rect = signupBannerTrigger.getBoundingClientRect(); // If page loaded up already scrolled below the trigger (via scroll restoration), show the banner by default if (rect.bottom < 0) { footerShown = true; signupBanner.classList.add('ds-signup-banner-visible'); } }); // Wait for trigger to fully enter viewport before showing banner (ensures PDF CTAs are never covered by banner) const observer = new IntersectionObserver((entries) => { entries.forEach(entry => { if (entry.isIntersecting && !footerShown) { footerShown = true; signupBanner.classList.add('ds-signup-banner-visible'); } else if (!entry.isIntersecting && footerShown) { if (signupBannerTrigger.getBoundingClientRect().bottom > 0) { footerShown = false; signupBanner.classList.remove('ds-signup-banner-visible'); } } }); }); observer.observe(signupBannerTrigger); // Set up signup banner close button event handler: const signupBannerCloseButton = document.querySelector('#ds-signup-banner-close-button'); signupBannerCloseButton.addEventListener('click', () => { signupBanner.classList.remove('ds-signup-banner-visible'); observer.unobserve(signupBannerTrigger); // Store the signup banner's visibility state in session storage sessionStorage.setItem('ds-signup-banner-hidden', 'true'); }); })();</script></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="96866615" data-landing_url="https://www.academia.edu/94399355/On_the_Index_of_Reducibility_of_Parameter_Ideals_the_Stable_and_Limit_Values" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="11153089" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/11153089/Index_of_reducibility_of_distinguished_parameter_ideals_and_sequentially_Cohen_Macaulay_modules">Index of reducibility of distinguished parameter ideals and sequentially Cohen-Macaulay modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="26926640" href="https://independent.academia.edu/Ho%C3%A0ngTr%C6%B0%E1%BB%9Dng4">Hoàng Trường</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Proceedings of the American Mathematical Society, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">It is shown that every sequentially Cohen-Macaulay module eventually has constant index of reducibility for distinguished parameter ideals.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Index of reducibility of distinguished parameter ideals and sequentially Cohen-Macaulay modules","attachmentId":46855939,"attachmentType":"pdf","work_url":"https://www.academia.edu/11153089/Index_of_reducibility_of_distinguished_parameter_ideals_and_sequentially_Cohen_Macaulay_modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/11153089/Index_of_reducibility_of_distinguished_parameter_ideals_and_sequentially_Cohen_Macaulay_modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="65445171" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/65445171/The_index_of_reducibility_of_parameter_ideals_in_low_dimension">The index of reducibility of parameter ideals in low dimension</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="52924997" href="https://independent.academia.edu/ShiroGoto">Shiro Goto</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Algebra, 2004</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we present results concerning the following question: If M is a finitely-generated module with finite local cohomologies over a Noetherian local ring (A, m), does there exist an integer ℓ such that every parameter ideal for M contained in m ℓ has the same index of reducibility? We show that the answer is yes if dim M = 1 or if dim M = 2 and depth M > 0. This research is closely related to work of Goto-Suzuki and Goto-Sakurai; Goto-Sakurai have supplied an answer of yes in case M is Buchsbaum.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The index of reducibility of parameter ideals in low dimension","attachmentId":77038902,"attachmentType":"pdf","work_url":"https://www.academia.edu/65445171/The_index_of_reducibility_of_parameter_ideals_in_low_dimension","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/65445171/The_index_of_reducibility_of_parameter_ideals_in_low_dimension"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="63698570" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/63698570/Eventually_index_of_reducibility_on_sequentially_Cohen_Macaulay_modules">Eventually index of reducibility on sequentially Cohen-Macaulay modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="114585135" href="https://independent.academia.edu/HleTruong">Hoang le Truong</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2015</p><p class="ds-related-work--abstract ds2-5-body-sm">It is shown that a module is sequentially Cohen-Macaulay if and only if the index of reducibility for distinguished parameter ideals are eventually constant with special value. As corollaries to the main theorem we given to characterize the Gorensteinness, Cohen-Macaulayness of local rings in term of eventually index of reducibility for distinguished parameter ideals.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Eventually index of reducibility on sequentially Cohen-Macaulay modules","attachmentId":76042102,"attachmentType":"pdf","work_url":"https://www.academia.edu/63698570/Eventually_index_of_reducibility_on_sequentially_Cohen_Macaulay_modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/63698570/Eventually_index_of_reducibility_on_sequentially_Cohen_Macaulay_modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="11153083" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/11153083/Asymptotic_behavior_of_parameter_ideals_in_generalized_Cohen_Macaulay_modules">Asymptotic behavior of parameter ideals in generalized Cohen–Macaulay modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="26926640" href="https://independent.academia.edu/Ho%C3%A0ngTr%C6%B0%E1%BB%9Dng4">Hoàng Trường</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Algebra, 2008</p><p class="ds-related-work--abstract ds2-5-body-sm">The purpose of this paper is to give affirmative answers to two open questions as follows. Let (R, m) be a generalized Cohen-Macaulay Noetherian local ring. Both questions, the first question was raised by M. Rogers and the second one is due to S. Goto and H. Sakurai [7], ask whether for every parameter ideal q contained in a high enough power of the maximal ideal m the following statements are true: (1) The index of reducibility NR(q; R) is independent of the choice of q; and (2) I 2 = qI, where I = q :R m.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Asymptotic behavior of parameter ideals in generalized Cohen–Macaulay modules","attachmentId":46855949,"attachmentType":"pdf","work_url":"https://www.academia.edu/11153083/Asymptotic_behavior_of_parameter_ideals_in_generalized_Cohen_Macaulay_modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/11153083/Asymptotic_behavior_of_parameter_ideals_in_generalized_Cohen_Macaulay_modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="11389877" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/11389877/On_the_uniform_bound_of_the_index_of_reducibility_of_parameter_ideals_of_a_module_whose_polynomial_type_is_at_most_one">On the uniform bound of the index of reducibility of parameter ideals of a module whose polynomial type is at most one</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="27698366" href="https://independent.academia.edu/QuyP">Pham Quy</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Archiv der Mathematik, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">On the uniform bound of the index of reducibility of parameter ideals of a module whose polynomial type is at most one 1</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the uniform bound of the index of reducibility of parameter ideals of a module whose polynomial type is at most one","attachmentId":46736797,"attachmentType":"pdf","work_url":"https://www.academia.edu/11389877/On_the_uniform_bound_of_the_index_of_reducibility_of_parameter_ideals_of_a_module_whose_polynomial_type_is_at_most_one","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/11389877/On_the_uniform_bound_of_the_index_of_reducibility_of_parameter_ideals_of_a_module_whose_polynomial_type_is_at_most_one"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="11153088" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/11153088/Parametric_decomposition_of_powers_of_parameter_ideals_and_sequentially_Cohen_Macaulay_modules">Parametric decomposition of powers of parameter ideals and sequentially Cohen-Macaulay modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="26926640" href="https://independent.academia.edu/Ho%C3%A0ngTr%C6%B0%E1%BB%9Dng4">Hoàng Trường</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Proceedings of the American Mathematical Society, 2008</p><p class="ds-related-work--abstract ds2-5-body-sm">Let M be a finitely generated module of dimension d over a Noetherian local ring (R, m) and q the parameter ideal generated by a system of parameters x = (x1, . . . , x d ) of M . For each positive integer n, set</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Parametric decomposition of powers of parameter ideals and sequentially Cohen-Macaulay modules","attachmentId":46855981,"attachmentType":"pdf","work_url":"https://www.academia.edu/11153088/Parametric_decomposition_of_powers_of_parameter_ideals_and_sequentially_Cohen_Macaulay_modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/11153088/Parametric_decomposition_of_powers_of_parameter_ideals_and_sequentially_Cohen_Macaulay_modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="63698569" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/63698569/On_the_index_of_reducibility_in_Noetherian_modules">On the index of reducibility in Noetherian modules</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="114585135" href="https://independent.academia.edu/HleTruong">Hoang le Truong</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Pure and Applied Algebra, 2015</p><p class="ds-related-work--abstract ds2-5-body-sm">Let M be a finitely generated module over a Noetherian ring R and N a submodule. The index of reducibility ir M (N) is the number of irreducible submodules that appear in an irredundant irreducible decomposition of N (this number is well defined by a classical result of Emmy Noether). Then the main results of this paper are: (1) ir M (N) = p∈Ass R (M/N) dim k(p) Soc(M/N) p ; (2) For an irredundant primary decomposition of N = Q 1 ∩ • • • ∩ Q n , where Q i is p i-primary, then ir M (N) = ir M (Q 1) + • • • + ir M (Q n) if and only if Q i is a p imaximal embedded component of N for all embedded associated prime ideals p i of N ; (3) For an ideal I of R there exists a polynomial Ir M,I (n) such that Ir M,I (n) = ir M (I n M) for n ≫ 0. Moreover, bight M (I) − 1 ≤ deg(Ir M,I (n)) ≤ ℓ M (I) − 1; (4) If (R, m) is local, M is Cohen-Macaulay if and only if there exist an integer l and a parameter ideal q of M contained in m l such that ir M (qM) = dim R/m Soc(H d m (M)), where d = dim M .</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the index of reducibility in Noetherian modules","attachmentId":76042179,"attachmentType":"pdf","work_url":"https://www.academia.edu/63698569/On_the_index_of_reducibility_in_Noetherian_modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/63698569/On_the_index_of_reducibility_in_Noetherian_modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="11153084" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/11153084/The_equality_in_sequentially_Cohen_Macaulay_rings">The equality in sequentially Cohen–Macaulay rings</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="26926640" href="https://independent.academia.edu/Ho%C3%A0ngTr%C6%B0%E1%BB%9Dng4">Hoàng Trường</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Algebra, 2013</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The equality in sequentially Cohen–Macaulay rings","attachmentId":46855969,"attachmentType":"pdf","work_url":"https://www.academia.edu/11153084/The_equality_in_sequentially_Cohen_Macaulay_rings","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/11153084/The_equality_in_sequentially_Cohen_Macaulay_rings"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="82647460" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/82647460/On_Hilbert_coefficients_and_sequentially_Cohen_Macaulay_rings">On Hilbert coefficients and sequentially Cohen-Macaulay rings</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="146437010" href="https://independent.academia.edu/YenHoang210">Yen Hoang</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Proceedings of the American Mathematical Society, 2021</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we explore the relation between the index of reducibility and the Hilbert coefficients in local rings. Consequently, the main result of this study provides a characterization of a sequentially Cohen-Macaulay ring in terms of its Hilbert coefficients of non-parameter ideals. As corollaries to the main theorem, we obtain characterizations of a Gorenstein/Cohen-Macaulay ring in terms of its Chern coefficients of non-parameter ideals.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On Hilbert coefficients and sequentially Cohen-Macaulay rings","attachmentId":88285686,"attachmentType":"pdf","work_url":"https://www.academia.edu/82647460/On_Hilbert_coefficients_and_sequentially_Cohen_Macaulay_rings","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/82647460/On_Hilbert_coefficients_and_sequentially_Cohen_Macaulay_rings"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="63698576" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/63698576/The_index_of_reducibility_of_powers_of_a_standard_parameter_ideal">The index of reducibility of powers of a standard parameter ideal</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="114585135" href="https://independent.academia.edu/HleTruong">Hoang le Truong</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Algebra and Its Applications</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper, we study the index of reducibility of powers of a standard parameter ideal. An explicit formula is proved for the extremal case. We apply the main result to compute Hilbert polynomials of socle ideals of standard parameter ideals.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The index of reducibility of powers of a standard parameter ideal","attachmentId":76042175,"attachmentType":"pdf","work_url":"https://www.academia.edu/63698576/The_index_of_reducibility_of_powers_of_a_standard_parameter_ideal","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/63698576/The_index_of_reducibility_of_powers_of_a_standard_parameter_ideal"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":96866615,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":96866615,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_96866615" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="91836519" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/91836519/On_sequentially_Cohen_Macaulay_modules">On sequentially Cohen-Macaulay modules</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="247711754" href="https://independent.academia.edu/DoanTrungCuong">Doan Trung Cuong</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Kodai Mathematical Journal, 2007</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On sequentially Cohen-Macaulay modules","attachmentId":95010130,"attachmentType":"pdf","work_url":"https://www.academia.edu/91836519/On_sequentially_Cohen_Macaulay_modules","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/91836519/On_sequentially_Cohen_Macaulay_modules"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="1" data-entity-id="16415667" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/16415667/Cohen_Macaulayness_versus_the_vanishing_of_the_first_Hilbert_coefficient_of_parameter_ideals">Cohen-Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="35562177" href="https://sauvage.academia.edu/ShiroGoto">Shiro Goto</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of the London Mathematical Society, 2010</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Cohen-Macaulayness versus the vanishing of the first Hilbert coefficient of parameter ideals","attachmentId":42492985,"attachmentType":"pdf","work_url":"https://www.academia.edu/16415667/Cohen_Macaulayness_versus_the_vanishing_of_the_first_Hilbert_coefficient_of_parameter_ideals","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/16415667/Cohen_Macaulayness_versus_the_vanishing_of_the_first_Hilbert_coefficient_of_parameter_ideals"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="2" data-entity-id="10015917" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/10015917/Rees_algebras_of_parameter_ideals">Rees algebras of parameter ideals</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="24384451" href="https://independent.academia.edu/JugalVerma">Jugal Verma</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Pure and Applied Algebra, 1989</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Rees algebras of parameter ideals","attachmentId":47568696,"attachmentType":"pdf","work_url":"https://www.academia.edu/10015917/Rees_algebras_of_parameter_ideals","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/10015917/Rees_algebras_of_parameter_ideals"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="3" data-entity-id="51139746" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/51139746/On_a_New_Invariant_of_Finitely_Generated_Modules_Over_Local_Rings">On a New Invariant of Finitely Generated Modules Over Local Rings</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="199496862" href="https://independent.academia.edu/NGUYENCUONG413">NGUYEN CUONG</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Algebra and Its Applications, 2010</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On a New Invariant of Finitely Generated Modules Over Local Rings","attachmentId":68980587,"attachmentType":"pdf","work_url":"https://www.academia.edu/51139746/On_a_New_Invariant_of_Finitely_Generated_Modules_Over_Local_Rings","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/51139746/On_a_New_Invariant_of_Finitely_Generated_Modules_Over_Local_Rings"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related topics</h2><div class="ds-research-interests--pills-container"><a class="js-related-research-interest ds-research-interests--pill" data-entity-id="300" rel="nofollow" href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a><a class="js-related-research-interest ds-research-interests--pill" data-entity-id="19997" rel="nofollow" href="https://www.academia.edu/Documents/in/Pure_Mathematics">Pure Mathematics</a><a class="js-related-research-interest ds-research-interests--pill" data-entity-id="2740863" rel="nofollow" href="https://www.academia.edu/Documents/in/Limit_Mathematics_">Limit (Mathematics)</a></div></div></div></div></div><div class="footer--content"><ul class="footer--main-links hide-on-mobile"><li><a href="https://www.academia.edu/about">About</a></li><li><a href="https://www.academia.edu/press">Press</a></li><li><a href="https://www.academia.edu/documents">Papers</a></li><li><a href="https://www.academia.edu/topics">Topics</a></li><li><a href="https://www.academia.edu/hiring"><svg style="width: 13px; height: 13px; position: relative; bottom: -1px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="briefcase" class="svg-inline--fa fa-briefcase fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M320 336c0 8.84-7.16 16-16 16h-96c-8.84 0-16-7.16-16-16v-48H0v144c0 25.6 22.4 48 48 48h416c25.6 0 48-22.4 48-48V288H320v48zm144-208h-80V80c0-25.6-22.4-48-48-48H176c-25.6 0-48 22.4-48 48v48H48c-25.6 0-48 22.4-48 48v80h512v-80c0-25.6-22.4-48-48-48zm-144 0H192V96h128v32z"></path></svg> <strong>We're Hiring!</strong></a></li><li><a href="https://support.academia.edu/hc/en-us"><svg style="width: 12px; height: 12px; position: relative; bottom: -1px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="question-circle" class="svg-inline--fa fa-question-circle fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M504 256c0 136.997-111.043 248-248 248S8 392.997 8 256C8 119.083 119.043 8 256 8s248 111.083 248 248zM262.655 90c-54.497 0-89.255 22.957-116.549 63.758-3.536 5.286-2.353 12.415 2.715 16.258l34.699 26.31c5.205 3.947 12.621 3.008 16.665-2.122 17.864-22.658 30.113-35.797 57.303-35.797 20.429 0 45.698 13.148 45.698 32.958 0 14.976-12.363 22.667-32.534 33.976C247.128 238.528 216 254.941 216 296v4c0 6.627 5.373 12 12 12h56c6.627 0 12-5.373 12-12v-1.333c0-28.462 83.186-29.647 83.186-106.667 0-58.002-60.165-102-116.531-102zM256 338c-25.365 0-46 20.635-46 46 0 25.364 20.635 46 46 46s46-20.636 46-46c0-25.365-20.635-46-46-46z"></path></svg> <strong>Help Center</strong></a></li></ul><ul class="footer--research-interests"><li>Find new research papers in:</li><li><a href="https://www.academia.edu/Documents/in/Physics">Physics</a></li><li><a href="https://www.academia.edu/Documents/in/Chemistry">Chemistry</a></li><li><a href="https://www.academia.edu/Documents/in/Biology">Biology</a></li><li><a href="https://www.academia.edu/Documents/in/Health_Sciences">Health Sciences</a></li><li><a href="https://www.academia.edu/Documents/in/Ecology">Ecology</a></li><li><a href="https://www.academia.edu/Documents/in/Earth_Sciences">Earth Sciences</a></li><li><a href="https://www.academia.edu/Documents/in/Cognitive_Science">Cognitive Science</a></li><li><a href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a></li><li><a href="https://www.academia.edu/Documents/in/Computer_Science">Computer Science</a></li></ul><ul class="footer--legal-links hide-on-mobile"><li><a href="https://www.academia.edu/terms">Terms</a></li><li><a href="https://www.academia.edu/privacy">Privacy</a></li><li><a href="https://www.academia.edu/copyright">Copyright</a></li><li>Academia ©2025</li></ul></div> </body> </html>