Geodesic
Prime and semiprime semigroup rings of cancellative semigroups
Glasgow mathematical journal, Tome 35 (1993) no. 1, pp. 1-12

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Let S be a cancellative semigroup. This paper is motivated by the problem of finding a description of semigroup rings K[S] over a field K that are semiprime or prime. Results of this type are well-known in the case of a group ring K[G], cf. [8]. The description, as well as the proofs, involve the FC-centre of G defined as the subset of all elements with finitely many conjugates in G. In [4], [5] Krempa extended the FC-centre techniques to the case of an arbitrary cancellative semigroup S. He defined a subsemigroup Δ(S) of S which coincides with the FC-centre in the case of groups, and can be used to describe the centre and to study special elements of K[S]. His results were strengthened by the author in [7], where Δ(S) was also applied in the context of prime and semiprime algebras K[S]. However, Δ(S) itself is not sufficient to characterize semigroup rings of this type. We note that in [2], [3] Dauns developed a similar idea for a study of the centre of semigroup rings and certain of their generalizations. 
Okniński, Jan. Prime and semiprime semigroup rings of cancellative semigroups. Glasgow mathematical journal, Tome 35 (1993) no. 1, pp. 1-12. doi: 10.1017/S0017089500009514
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