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Group Theory</a> <a href="/?q=in%3A514727" title="Articles in this Issue">27, No. 4, 789-811 (2024)</a>. </div> <div class="abstract">Three classical conjectures by Kaplansky state that if \(G\) is a torsion free group and \(K\) a field, then the group ring \(KG\) does neither contain a non-trivial unit nor zero-divisor nor non-trivial idempotent. Here a trivial unit is an element of shape \(kg\), with \(k \in K^\times\) and \(g \in G\), and the trivial idempotents are \(0\) and \(1\). While the conjecture on units has been recently disproved [<span class="zbmathjax-textit">G. Gardam</span>, Ann. Math. (2) 194, No. 3, 967–979 (2021; <a href="/1494.16026">Zbl 1494.16026</a>)] (results in odd characteristic and characteristic \(0\) are preprints at the time of writing), the other two remain open.<br class="zbmathjax-paragraph">In this paper the author proposes a generalization of the conjectures to rings graded by torsion free groups, a structure which appears naturally in many areas of mathematics and includes, e.g., crossed products. Some weak additional assumptions are needed to make Kaplansjy’s question interesting in this context: namely the ring should be unital, the grading should be non-degenerate and the identity component of the ring should be a domain. With this prerequisites the author shows that some of the classical results about Kaplansky’s conjectures also hold in this broader setting. Among others, he shows that they hold when the graded ring \(R\) is commutative or the group grading \(R\) is unique product or when the unit/zero divisor/idempotent in question is central. It is proven that the hierarchy of Kaplansky’s conjectures also carries over, i.e., the unit conjecture implies the zero divisor conjecture and it implies in turn the idempotent conjecture also in the setting studied here.<br class="zbmathjax-paragraph">Many examples are also provided, which on one hand show that the assumptions made are necessary but also that the results are really stronger than those known for group rings. The author also conjectures that his generalized formulations should hold true when the identity component of the graded ring has characteristic \(0\). We remark that the unit conjecture has been disproved since this paper appeared also over big enough rings of characteristic \(0\) in a paper by Gardam [loc. cit.].<div class="reviewer"> Reviewer: <a href="/authors/?q=rv%3A18563">Leo Margolis (Madrid)</a></div> <div class="clearfix"></div></div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7874819">Cited in <strong>2</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A16S34" title="MSC2020">16S34</a> </td> <td class="space"> Group rings </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A20K15" title="MSC2020">20K15</a> </td> <td class="space"> Torsion-free groups, finite rank </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AKaplansky+conjecture">Kaplansky conjecture</a>; <a href="/?q=ut%3Agraded+rings">graded rings</a>; <a href="/?q=ut%3Atorsion+free+group">torsion free group</a>; <a href="/?q=ut%3Aunit+conjecture">unit conjecture</a>; <a href="/?q=ut%3Azero+divisor+conjecture">zero divisor conjecture</a>; <a href="/?q=ut%3Aidempotent+conjecture">idempotent conjecture</a></div><div class="keywords"> <h3>Citations:</h3><a href="/1494.16026">Zbl 1494.16026</a></div> <!-- Modal used to show zbmath metadata in different output formats--> <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;"> <!-- alert for connection errors etc --> </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"> <!-- Button trigger metadata modal --> <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1550.16027" data-ciurl="/ci/07874819" data-biburl="/bibtex/07874819.bib" data-amsurl="/amsrefs/07874819.bib" data-xmlurl="/xml/07874819.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07874819.pdf" title="Zbl 1550.16027 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/10.1515/jgth-2023-0110" aria-label="DOI for “Units, zero-divisors and idempotents in rings graded by torsion-free groups”" title="10.1515/jgth-2023-0110">DOI</a> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/1904.04847"title="Note: arXiv document may differ from published version">arXiv</a> </div> <div class="sfx" style="float: right;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" title="Open Access License" class="cc-license-link no-new-tab-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/cc.svg" alt="Creative Commons CC license icon" class="cc-license-icon"> <img src="https://static.zbmath.org/contrib/img/cc/svg/icons/by.svg" alt="Creative Commons BY license icon" class="cc-license-icon"> </a> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">G. 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