On a theorem of Cohen and Montgomery for graded rings (Q2771446)

It was proved by \textit{M. Cohen} and \textit{S. Montgomery} [Trans. Am. Math. Soc. 282, 237-258 (1984; Zbl 0533.16001)] that \(J(R_e)=R_e\cap J(R)\) for any group graded ring \(R\) (where \(e\) is the identity element of the grading group). The author proves that for a semigroup \(S\) we have that \(J(R_e)=R_e\cap J(R)\) for any \(S\)-graded ring \(R\) and any idempotent \(e\in S\) if and only if either \(S\) has no idempotents or \(S\) has a minimal ideal which is a locally finite group. Also, for a semigroup \(S\) with zero, \(J(R_e)=R_e\cap J(R)\) for any contracted \(S\)-graded ring \(R\) and any idempotent \(e\in S\) if and only if the ideal generated by all idempotents of \(S\) is a direct union of locally finite groups.