A note on semiperfect group rings (Q1318905)

Let \(R\) be a commutative ring with identity and let \(G\) be an abelian group. Denote by \(J(R)\) the Jacobson radical of \(R\). \(R\) is said to be semiperfect if \(R/J(R)\) is Artinian and the idempotents of \(R/J(R)\) can be lifted to \(R\). It is proved that the group ring \(RG\) is semiperfect if and only if (1) \(R\) is semiperfect; (2) idempotents of \(RG/J(R)G\) can be lifted to \(RG\); (3) \(G\) is finite, or \(G \cong G_ p \times H\) where \(G_ p\) is an infinite \(p\)-group, \(H\) is finite, \(p \nmid | H|\) and \(R/J(R)\) is of characteristic \(p\) for some prime \(p\).