On a theorem of Cohen and Montgomery for graded rings

Giving as answer to Bergman's question, Cohen and Montgomery proved that, f or every finite group G with identity e and each G-graded ring R = circle t imes (g is an element ofG) R-g, the Jacobson radical J(R-e) of the initial component R-e is equal to R-e boolean AND J(R). We describe all semigroups S, which satisfy the following natural analogue of this property: J(R-e) = R-e boolean AND J(R) for each S-graded ring R = circle times (s is an eleme nt ofS) R-s and every idempotent e is an element of S.