Local contracted semigroup rings

The local contracted semigroup rings $R_0S$ over non-radical rings $R$ ($\overline R=R/J(R)\ne\{0\}$) are under consideration. The following main statement is proved. Let $R$ be a ring, $\overline R\ne\{0\}$, $S$ be a semigroup with zero. The ring $R_0S$ is local if and only if: (i) there exists a nil ideal $N\subseteq S$ such that $S/N\cong T^0$ is a semigroup $T$ (without zero) with adjoint zero; (ii) $RT$ is local, $R_0N$ is radical.