Units, zero-divisors and idempotents in rings graded by torsion-free groups (Q6565829)

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 [\textit{G. Gardam}, Ann. Math. (2) 194, No. 3, 967--979 (2021; Zbl 1494.16026)] (results in odd characteristic and characteristic \(0\) are preprints at the time of writing), the other two remain open.\N\NIn 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.\N\NMany 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.].