Centrally essential group algebras (Q1655800)

A ring \(R\) with centre \(C\) is said to be \textit{centrally essential} if the module \(R_C\) is an essential extension of the module \(C_C\). In the present paper the authors investigate group rings over fields which are centrally essential. The main result they prove is that if \(F\) is a field of characteristic \(p>0\) and \(G\) is a finite group, then the group algebra \(FG\) is centrally essential if, and only if, \(G=P\times H\) where \(P\) is the unique \(p\)-Sylow of \(G\), \(H\) is abelian and \(FP\) is centrally essential. The case of centrally essential modular group rings of finite \(p\)-groups is also studied (as recalled in Remark 1.2, in the characteristic zero case, for any group \(G\), \(FG\) is centrally essential if, and only if, \(G\) is abelian).