Nilpotent semigroups and semigroup algebras
First, the structure of nilpotent semigroups is discussed. If \(S\) is a completely 0-simple semigroup over a maximal group \(G\), then \(S\) is nilpotent if and only if \(G\) is nilpotent and \(S\) is an inverse semigroup. The main results on semigroup algebras are very interesting, but technical; they examine the prime homomorphic images of semigroup algebras \(K[S]\) of nilpotent semigroups. The classical ring of quotients of such a prime Goldie ring \(K[S]/ P\) actually comes from a prime homomorphic image of the group algebra \(K[G]\) of a nilpotent group. The prime radical of \(K[S]\) and the congruence it determines on \(S\) are studied. A full description is given for nilpotent semigroups of index 2; the prime ideals of these semigroups are completely prime. The last section of the paper contains many interesting examples that illustrate the uses of nilpotent semigroups and their semigroup algebras.
scientific article; zbMATH DE number 701290