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Firstly, we answer the question about elements which are always contained in the limit closure of a system of parameters. Then we apply it to give a characterization of systems of parameters which is a generalization of previous results of Dutta and Roberts in [11] and of Fouli and Huneke in [12]. We also prove a topological characterization of unmixed local rings. In two dimensional case, we compute explicitly the limit closure of a system of parameters. 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We consider conditions for these rings to have Cohen-Macaulay formal fibers. We also present several examples illustrating these results.</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":"Intermediate rings between a local domain and its completion. {II}","attachmentId":44830751,"attachmentType":"pdf","work_url":"https://www.academia.edu/24498380/Intermediate_rings_between_a_local_domain_and_its_completion_II_","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/24498380/Intermediate_rings_between_a_local_domain_and_its_completion_II_"><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="24498377" 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/24498377/Integral_closures_of_ideals_in_completions_of_regular_local_domains">Integral closures of ideals in completions of regular local domains</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="35338296" href="https://purdue.academia.edu/WilliamHeinzer">William Heinzer</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Lecture Notes in Pure and Applied Mathematics, 2005</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":"Integral closures of ideals in completions of regular local domains","attachmentId":44830738,"attachmentType":"pdf","work_url":"https://www.academia.edu/24498377/Integral_closures_of_ideals_in_completions_of_regular_local_domains","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/24498377/Integral_closures_of_ideals_in_completions_of_regular_local_domains"><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="67945940" 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/67945940/Zariski_Lipman_Theory_of_Complete_Ideals_in_2_DIMENSIONAL_Regular_Local_Rings">Zariski-Lipman Theory of Complete Ideals in 2-DIMENSIONAL Regular Local 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="24384451" href="https://independent.academia.edu/JugalVerma">Jugal Verma</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2003</p><p class="ds-related-work--abstract ds2-5-body-sm">The objective of these notes is to present a few important results about complete ideals in two-dimensional regular local rings. The fundamental theorems about such ideals are due to Zariski found in appendix 5 of [16]. These results were proved by Zariski in [17] for two dimensional polynomial rings over an algebraically closed field of characteristic zero. Zariski states in [17], “It is the main purpose of the present investigation to develop an arithmetic theory parallel to the geometric theory of infinitely near points (in plane or on a surface without singularities).” In order to state Zariski’s results, we recall the notion of integral closure of an ideal.</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":"Zariski-Lipman Theory of Complete Ideals in 2-DIMENSIONAL Regular Local Rings","attachmentId":78602821,"attachmentType":"pdf","work_url":"https://www.academia.edu/67945940/Zariski_Lipman_Theory_of_Complete_Ideals_in_2_DIMENSIONAL_Regular_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-wsj-grid-card-view-pdf" href="https://www.academia.edu/67945940/Zariski_Lipman_Theory_of_Complete_Ideals_in_2_DIMENSIONAL_Regular_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><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":96866625,"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":96866625,"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_96866625" 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. 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