On isomorphism of group algebras of torsion Abelian groups (Q1209566)

Suppose \(R\) is a commutative ring with 1, \(G\) is a torsion Abelian group and \(RG\) is the group algebra of \(G\) over \(R\). Let \(\text{inv}(R)\) be the set of rational primes \(p\) which invert in \(R\) and set \(G_ R=\coprod\{G_ p:p\in\text{inv}(R)\}\). The group \(G\) is called \(R\)- favorable if \(G_ R\) is the trivial subgroup of \(G\). It is proved that the following problems are equivalent: (1) For every field \(F\) of nonzero characteristic \(p\) and for every \(p\)-group \(G\), \(FG\cong FH\) for some group \(H\) implies that \(G\cong H\). (2) For every ring \(R\) of characteristic 0 and for all \(R\)-favorable torsion groups \(G\) and \(H\), \(RG\cong RH\) implies that \(G\cong H\). If the additive group of \(R\) is assumed to be torsion free and if \(G\) is \(R\)-favorable it is shown that the isomorphism class of \(RG\) determines the isomorphism class of \(G\).