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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)

##### MSC:
 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16S34 Group rings 20K10 Torsion groups, primary groups and generalized primary groups
##### Keywords:
isomorphism problem; group algebras; Abelian $$p$$-groups
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