Geodesic
Radical Semigroup Rings and the Thue--Morse Semigroup
Matematičeskie zametki, Tome 74 (2003) no. 4, pp. 529-537

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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. 
@article{MZM_2003_74_4_a5,
     author = {I. B. Kozhukhov},
     title = {Radical {Semigroup} {Rings} and the {Thue--Morse} {Semigroup}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {529--537},
     publisher = {mathdoc},
     volume = {74},
     number = {4},
     year = {2003},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2003_74_4_a5/}
}
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I. B. Kozhukhov. Radical Semigroup Rings and the Thue--Morse Semigroup. Matematičeskie zametki, Tome 74 (2003) no. 4, pp. 529-537. http://geodesic.mathdoc.fr/item/MZM_2003_74_4_a5/