commutative group algebras; isomorphism
Suppose $p$ is a prime number and $R$ is a commutative ring with unity of characteristic 0 in which $p$ is not a unit. Assume that $G$ and $H$ are $p$-primary abelian groups such that the respective group algebras $RG$ and $RH$ are $R$-isomorphic. Under certain restrictions on the ideal structure of $R$, it is shown that $G$ and $H$ are isomorphic.
[U] Ullery W.: On isomorphism of group algebras of torsion abelian groups
. Rocky Mtn. J. Math. 22 (1992), 1111-1122. MR 1183707
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