A ring means an associative ring (not necessarily with unity). Let \(R\) be a ring, \(A\subseteq R\). The set \(C_R(A)=\{r\in R\mid ar=ra\) for all \(a\in A\}\) is called the centralizer of \(A\) in \(R\). Let \(S\) be a semigroup, and let \(R[S]\) be the semigroup ring of \(S\) over \(R\). Define \(S'=\{x\in S\mid xa=xb\) or \(ax=bx\) implies \(a=b\) for all \(a,b\in S\}\). For a ring \(R\) let \(\text{Ann}_l(R)=\{x\in R\mid xR=0\}\), \(\text{Ann}_r(R)=\{x\in R\mid Rx=0\}\). Suppose that either \(\text{Ann}_l(R)=0\) or \(\text{Ann}_r(R)=0\). Under these conditions the main results (Theorems A and B) are proved. Theorem A gives a description of \(C_R(M)\) for any non-empty \(M\subseteq S'\). Theorem B states that, in the case \(S'\neq\emptyset\) and \(S\setminus S'\) is a commutative ideal of \(S\), the supporting subsemigroup of any \(e=e^2\in C_{R[S]}(R[S'])\) is finite. According to Remark 2.4, Theorem B remains true if one changes the condition ``\(S\setminus S'\) is a commutative ideal of \(S\)'' to ``\(S\setminus S'\) is either empty or a commutative ideal of \(S\)''. With that Theorem B may be considered as a generalization of a well known theorem; the supporting subgroup of a central idempotent in a group ring is finite.