On semilocal semigroup rings

The approach for study semilocal semigroup rings over non-radical rings based on description the structure of semigroup on the whole is suggested. The following main statement is proved. Let $R$ be a ring, $\overline R=R/J(R)\ne 0$, $S$ be a semigroup with zero $z$. The semigroup ring $RS$ is semilocal if and only if: $(i)$ $R$ is semilocal; $(ii)$ there exists a chain of ideals $\{z\}=S_0\subset S_1\subset\ldots\subset S_n=S$ such that $S_i/S_{i-1}$, $1\le i\le n$, are nil or completely $0$-simple; $(iii)$ the contracted semigroup rings $R_0(S_i/S_{i-1})$, are semilocal.