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A note on group algebras of \(p\)-primary abelian groups. (English) Zbl 0828.20005
Let \(RG\) be the group algebra of an abelian \(p\)-group \(G\) over a commutative ring \(R\) with identity of characteristic 0. We let \(Rp\) denote the ideal of \(R\) generated by \(p\) and \(J(R)\) the Jacobson radical of \(R\). Let \(zd(R)\) be the set of prime numbers that are zero divisors in \(R\) and let \(\text{inv} (R)\) be the set of prime numbers that are units in \(R\). The following result is established. Theorem. If \(G\) and \(H\) are abelian \(p\)-groups such that \(RG \cong RH\) as \(R\)-algebras and \(p \notin \text{inv} (R)\) then \(G \cong H\) in each of the following cases: (1) \(Rp\) contains no nonzero idempotents. (2) \(p \in J(R)\). (3) \(p \notin zd(R)\).
Reviewer: T.Mollov (Plovdiv)

20C07 Group rings of infinite groups and their modules (group-theoretic aspects)
16S34 Group rings
20K10 Torsion groups, primary groups and generalized primary groups
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