The units and idempotents in the group ring \(K(\mathbb{Z}_m\times\mathbb{Z}_n)\) (Q2764907)

Let \(G\) be the direct product of two finite cyclic groups and \(K\) an algebraically closed field of characteristic 0. In terms of values of certain polynomials criteria are given for an element of the group ring \(KG\) be a unit or an idempotent. It is also proved that the trace, i.e. the coefficient of the identity group element, of an idempotent is one of the rationals: \(0\), \(1/|G|\), \(2/|G|\),\dots, \((|G|-1)/|G|\) or \(1\).NEWLINENEWLINENEWLINEThere are inaccuracies in the formulation of some statements. In particular, the author says, referring to the article by \textit{G. H. Cliff} and \textit{S. K. Sehgal} [published in Proc. Am. Math. Soc. 62, 11-14 (1977; Zbl 0347.16007)], that the authors proved that if \(\alpha\) is an idemponent of the group algebra of a polycyclic-by-finite group over a field of characteristic \(0\) then the trace \(\text{tr}(\alpha)\) of \(\alpha\) is of the form \(r/s\) with \((r,s)=1\). But this says only that \(\text{tr}(\alpha)\) is a rational number, which is known by an earlier result of A. Zalesskij. As a matter of fact, G. H. Cliff and S. K. Sehgal proved that if a prime \(p\) divides \(s\) then the support of \(\alpha\) contains a non-trivial \(p\)-element, at which the partial augmentation of \(\alpha\) does not vanish. Another inaccuracy is that the author formulates Theorem 1 (and Theorem 2) for a direct product of two cyclic groups of relatively prime orders, however, the reader should omit this restriction on the orders.