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Commutative group algebras generated by idempotents. (English) Zbl 1100.16022

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).

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
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