Radical Semigroup Rings and the Thue–Morse Semigroup

Let $R$ be an associative ring with unit, let $S$ be a semigroup with zero, and let $RS$ be a contracted semigroup ring. It is proved that if $RS$ is radical in the sense of Jacobson and if the element 1 has infinite additive order, then $S$ is a locally finite nilsemigroup. Further, for any semigroup $S$, there is a semigroup $T\supset S$ such that the ring $RT$ is radical in the Brown–McCoy sense. Let $S$ be the semigroup of subwords of the sequence $abbabaabbaababbab...$, and let $F$ be the two-element field. Then the ring $FS$ is radical in the Brown–McCoy sense and semisimple in the Jacobson sense.