Document Zbl 1454.16043

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Algebra Appl.</a> <a href="/?q=in%3A448802" title="Articles in this Issue">19, No. 2, Article ID 2050029, 9 p. (2020)</a>. </div> <div class="abstract">Summary: An element $a$ in a ring $R$ has a gs-Drazin inverse if there exists $p^2 = p \in \mathrm{comm}^2(a)$ such that $a - p \in R^{\mathrm{qnil}}$. In this paper, we extend Cline’s formula and Jacobson’s lemma for gs-Drazin inverses. Various additive properties of gs-Drazin inverses are thereby obtained.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7195860">Cited in <strong>1</strong> Document</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A16U90" title="MSC2020">16U90</a> </td> <td class="space"> Generalized inverses (associative rings and algebras) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A16E50" title="MSC2020">16E50</a> </td> <td class="space"> von Neumann regular rings and generalizations (associative algebraic aspects) </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Ags-Drazin+inverse">gs-Drazin inverse</a>; <a href="/?q=ut%3ACline%27s+formula">Cline’s formula</a>; <a href="/?q=ut%3AJacobson%27s+lemma">Jacobson’s lemma</a>; <a href="/?q=ut%3Aadditive+property">additive property</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1454.16043" data-ciurl="/ci/07195860" data-biburl="/bibtex/07195860.bib" data-amsurl="/amsrefs/07195860.bib" data-xmlurl="/xml/07195860.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07195860.pdf" title="Zbl 1454.16043 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1142/S0219498820500292" aria-label="DOI for “On gs-Drazin inverses in a ring”" title="10.1142/S0219498820500292">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Cvetkovic-Ilic, D. S., The generalized Drazin inverse with commutativity up to a factor in a Banach algebra, Linear Algebra Appl.431 (2009) 783-791. · <a href="/1214.47005" class="nowrap">Zbl 1214.47005</a></td> </tr><tr> <td>[2]</td> <td class="space">Cvetkovic-Ilic, D. S. and Deng, C., Some results on the Drazin invertibility and idempotents, J. Math. Anal. Appl.359 (2009) 731-738. · <a href="/1181.46035" class="nowrap">Zbl 1181.46035</a></td> </tr><tr> <td>[3]</td> <td class="space">Cvetkovic-Ilic, D. S., Djordjevic, D. S. and Wei, Y., Additive results for the generalized Drazin inverse in a Banach algebra, Linear Algebra Appl.418 (2016) 53-61. · <a href="/1104.47040" class="nowrap">Zbl 1104.47040</a></td> </tr><tr> <td>[4]</td> <td class="space">Cvetkovic-Ilic, D. and Harte, R., On Jacobson’s lemma and Drazin invertibility, Appl. Math. Lett.23 (2010) 417-420. · <a href="/1195.16033" class="nowrap">Zbl 1195.16033</a></td> </tr><tr> <td>[5]</td> <td class="space">Djordjevic, D. S. and Wei, Y., Additive results for the generalized Drazin inverse, J. Aust. Math. Soc.73 (2002) 115-125. · <a href="/1020.47001" class="nowrap">Zbl 1020.47001</a></td> </tr><tr> <td>[6]</td> <td class="space">Gurgun, O., Properties of generalized strongly Drazin invertible elements in general rings, J. Algebra Appl.16 (2017), Article ID:1750207, 13 pp., https://doi.org/10.1142/S0219498817502073. · <a href="/1392.16035" class="nowrap">Zbl 1392.16035</a></td> </tr><tr> <td>[7]</td> <td class="space">Harte, R., Invertibility and Singularity for Bounded Linear Operators (Marcel Dekker, New York, 1988). · <a href="/0636.47001" class="nowrap">Zbl 0636.47001</a></td> </tr><tr> <td>[8]</td> <td class="space">Koliha, J. J., A generalized Drazin inverse, Glasgow Math. J.38 (1996) 367-381. · <a href="/0897.47002" class="nowrap">Zbl 0897.47002</a></td> </tr><tr> <td>[9]</td> <td class="space">Koliha, J. J., Cvetkovic-Ilic, D. S. and Deng, C., Generalized Drazin invertibility of combinations of idempotents, Linear Algebra Appl.437 (2012) 2317-2324. · <a href="/1259.46038" class="nowrap">Zbl 1259.46038</a></td> </tr><tr> <td>[10]</td> <td class="space">Liao, Y., Chen, J. and Cui, J., Cline’s formula for the generalized Drazin inverse, Bull. Malays. Math. Sci. Soc.37 (2014) 37-42. · <a href="/1298.15009" class="nowrap">Zbl 1298.15009</a></td> </tr><tr> <td>[11]</td> <td class="space">Mosic, D., Reverse order laws for the generalized strongly Drazin inverses, Appl. Math. Comp.284 (2016) 37-46. · <a href="/1410.46030" class="nowrap">Zbl 1410.46030</a></td> </tr><tr> <td>[12]</td> <td class="space">Mosic, D., Extensions of Jacobson’s lemma for Drazin inverses, Aequat. Math.91(3) (2017) 419-428. · <a href="/1388.16038" class="nowrap">Zbl 1388.16038</a></td> </tr><tr> <td>[13]</td> <td class="space">Zeng, Q. and Zhong, H., Common properties of bounded linear operators $A C$ and $B A$: local spectral theory, J. Math. Anal. Appl.414 (2014) 553-560. · <a href="/1308.47002" class="nowrap">Zbl 1308.47002</a></td> </tr><tr> <td>[14]</td> <td class="space">Zeng, Q. and Zhong, H., New results on common properties of the products $A C$ and $B A$, J. Math. Anal. Appl.427 (2015) 830-840. · <a href="/1327.47001" class="nowrap">Zbl 1327.47001</a></td> </tr><tr> <td>[15]</td> <td class="space">Zhuang, G., Chen, J. and Cui, J., Jacobson’s lemma for the generalized Drazin inverse, Linear Algebra Appl.436 (2012) 742-746. · <a href="/1231.15008" class="nowrap">Zbl 1231.15008</a></td> </tr><tr> <td>[16]</td> <td class="space">Zhuang, G., Chen, J., Cvetkovic-Ilic, D. S. and Wei, Y., Additive property of drazin invertibility of elements in a ring, Linear Mult. Algebra60 (2012) 903-910. · <a href="/1261.15009" class="nowrap">Zbl 1261.15009</a></td> </tr><tr> <td>[17]</td> <td class="space">Zeng, Q., Wu, Z. and Wen, Y., New extensions of Cline’s formula for generalized inverses, Filomat31 (2017) 1973-1980. · <a href="/1484.47004" class="nowrap">Zbl 1484.47004</a></td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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