Document Zbl 1262.47002

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An element $a \in R$ is called pseudopolar if there exists an idempotent $p$ in the double commutant of $\{a\}$ such that $a + p \in U(R)$ and $a^kp \in J(R)$ for some positive integer $k$. The ring $R$ is said to be pseudopolar if each element of $R$ is pseudopolar. The idempotent $p$ satisfying the above conditions is called a strongly spectral idempotent of $a$. The authors show that the strongly spectral idempotent is unique and correspond the uniqueness of the strongly spectral idempotent to the double commutant. They connect pseudopolar rings with strongly $\pi$-regular rings, semiregular rings and uniquely strongly clean rings.<br class="zbmathjax-paragraph">An element $a \in R$ is called pseudo Drazin invertible if there exists an element $b$ in the double commutant of $\{a\}$ such that $ab^2 = b$ and $a^k - a^{k+1}b \in J(R)$ for some positive integer $k$. The authors prove that an element is pseudopolar if and only if it is pseudo Drazin invertible. They also extend some basic properties of Drazin inverses to pseudo Drazin inverses.<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A10123">Mohammad Sal Moslehian (Mashhad)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A6063135">Cited in <strong>3</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A6063135">Cited in <strong>39</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A47A05" title="MSC2020">47A05</a> </td> <td class="space"> General (adjoints, conjugates, products, inverses, domains, ranges, etc.) </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><tr> <td> <a class="mono" href="/classification/?q=cc%3A16S99" title="MSC2020">16S99</a> </td> <td class="space"> Associative rings and algebras arising under various constructions </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A15A09" title="MSC2020">15A09</a> </td> <td class="space"> Theory of matrix inversion and generalized inverses </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Apseudopolar+rings">pseudopolar rings</a>; <a href="/?q=ut%3Astrongly+%5C%28%5Cpi%5C%29-regular+rings">strongly $\pi$-regular rings</a>; <a href="/?q=ut%3Apseudo+Drazin+inverses">pseudo Drazin inverses</a>; <a href="/?q=ut%3Auniquely+strongly+clean+rings">uniquely strongly clean rings</a>; <a href="/?q=ut%3ABanach+algebras">Banach algebras</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 1262.47002" data-ciurl="/ci/06063135" data-biburl="/bibtex/06063135.bib" data-amsurl="/amsrefs/06063135.bib" data-xmlurl="/xml/06063135.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/06063135.pdf" title="Zbl 1262.47002 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/10.1016/j.laa.2012.04.039" aria-label="DOI for “Pseudo Drazin inverses in associative rings and Banach algebras”" title="10.1016/j.laa.2012.04.039">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">Anderson, D. 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