Commutative group algebras whose quotient rings by nilradicals are generated by idempotents. (Q627383)

Let \(G\) be an Abelian group and \(R\) a commutative ring with identity. Denote by \(RG\) the group algebra of \(G\) over \(R\) and by \(\text{nil}(RG)\) the nil radical of \(RG\). The authors give necessary and sufficient conditions \(RG/\text{nil}(RG)\) to be generated by idempotents over \(R\) (Theorem 2.12). They prove (Theorem 2.18) that the following conditions are equivalent: (i) \(RG\) is generated by idempotents over \(R\) and (ii) \((R/\text{nil}(R))G\) is generated by idempotents over \(R/\text{nil}(R)\). The authors establish (Theorem 2.19) that if the ring \(R\) is indecomposable and \(\text{char}(R)=0\), then the condition (i) is equivalent to the following condition: (iii) \(RG/\text{nil}(RG)\) is generated by idempotents over \(R\).