The converse of Schur's lemma in group rings. (Q2458375)

The paper under review explores the converse of Schur's lemma (CSL). The ring \(R\) is said to be a CLS-ring if every module is simple whenever its endomorphism ring is a division ring. Let \(M\) be an Abelian group and let \(G\) be a group of automorphisms of \(M\). For every subgroup \(L\) of \(M\) we denote \(L^G=\{x\in L\mid\sigma(x)=x,\;\forall x\in G\}\). The subgroup \(L\) is said to be stable by \(G\) (\(G\)-stable) if \(\sigma(x)=x\) for every \(x\in L\) and every \(\sigma\in G\). For every positive integer \(n\) we denote \(T_n(M)=\{x\in M\mid nx=0\}\) the \(n\)-torsion subgroup of \(M\). For a finite group \(G\) and a prime \(p\) we say that \(g\in G\) is a \(p'\)-element if the order of \(g\) is prime to \(p\). The following results are obtained. Theorem 4. Let \(R\) be a semiprime ring, \(G\) a finite group of automorphisms of \(R\). Assume that for every prime integer \(p\) for which \(T_p(R)\neq 0\), the set of \(p'\)-elements of \(G\) is a subgroup of \(G\). Then for every \(G\)-stable nonzero left or right ideal \(I\) we have \(I^G\neq 0\). Theorem 6. Let \(A\) be a commutative and perfect ring, \(G\) a finite group. The following assertions are equivalent: (i) \(A[G]\) is a primary decomposable ring. (ii) \(A[G]\) is a CSL-ring. (iii) For each prime number \(p\) such that \(T_p(A)\neq 0\), there exists a \(p'\)-subgroup \(H\) of \(G\), and a \(p\)-Sylow subgroup \(P\) of \(G\) such that \(G=HP\).