Centralizers and Central Idempotents of Semigroup Rings

Let A be a nonempty subset of an associative ring R . Call the subring CR(A)={r∈ R\mid ra=ar \quadfor all\quad a∈ A} of R the centralizer of A in R . Let S be a semigroup. Then the subsemigroup S'= {s∈ S\mid sa=sb \quador\quad as=bs \quadimplies\quad a=b \quadfor all a,b∈ S} of S is called the C -subsemigroup. In this paper, the centralizer CR[S](R[M]) for the semigroup ring R[S] will be described, where M is any nonempty subset of S' . An non-zero idempotent e is called the central idempotent of R[S] if e lies in the center of R[S] . Assume that S\backslash S' is a commutative ideal of S and Annl(R)=0 . Then we show that the supporting subsemigroup of any central idempotent of R[S] must be finite.