Kawai, Hideyasu; Onoda, Nobuharu Commutative group algebras generated by idempotents. (English) Zbl 1100.16022 Toyama Math. J. 28, 41-54 (2005). Let \(RG\) be the group algebra of an Abelian group \(G\) over a commutative ring \(R\) with identity. In this paper, necessary and sufficient conditions are given for (i) a finitely generated \(R\)-algebra \(A\) to be generated by idempotents over \(R\) when \(A\) is a projective \(R\)-module (Theorem 2.3) and (ii) the group algebra \(RG\) to be generated by idempotents (Theorem 4.2). As a corollary it is obtained that the group algebra \(RG\) of a finite Abelian group \(G\) of order \(n\) is generated by idempotents if and only if \(RG\) is isomorphic to the direct product of \(n\) copies of \(R\) as an \(R\)-algebra (Corollary 4.4). Reviewer: Todor Mollov (Plovdiv) Cited in 3 Documents MSC: 16S34 Group rings 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting) 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20K10 Torsion groups, primary groups and generalized primary groups Keywords:commutative group algebras; projective Abelian \(p\)-groups; isomorphism problem; direct factor problem; direct products PDFBibTeX XMLCite \textit{H. Kawai} and \textit{N. Onoda}, Toyama Math. J. 28, 41--54 (2005; Zbl 1100.16022)