Local semigroup rings

The description of local semigroup algebras over a field of characteristic $p$ (if $p>0$, then semigroups are assumed to be locally finite) due to J. Okninsky (1984) is transferred to semigroup rings over non-radical rings. The following statement is proved. Let $R$ be a ring, $R\ne J(R)$, $\operatorname{char}R=0$ ($\operatorname{char}R=p>1$), $S$ be a semigroup (respectively, a locally finite semigroup). The semigroup ring $R[S]$ is local [scalar local] if and only if $R$ is such a ring and $S$ is an ideal extension of a rectangular band (respectively of a completely simple semigroup over a $p$-group) by a locally nilpotent semigroup.