On permutative semigroup algebras
Let F be a field and let S be a permutative semigroup, that is a semigroup with a non-trivial permutation identity \(x_ 1x_ 2...x_ n=x_{\sigma (1)}x_{\sigma (2)}...x_{\sigma (n)}\). It is shown that the Jacobson radical of the semigroup algebra F[S] is a nil ideal and it coincides with the subspace of F[S] spanned by the set \(\{\) s-t\(|\) (s,t)\(\in \nu \}\) for a congruence \(\nu\) on S. Here \(\nu\) is the least separative congruence on S if \(ch(F)=0\), and ``the least p-separative'' congruence if \(ch(F)=p>0\). The second main result establishes a criterion for the semisimplicity of a PI semigroup algebra of an arbitrary separative semigroup. These results extend the description of the radical of commutative semigroup algebras due to \textit{W. D. Munn} [Math. Proc. Camb. Philos. Soc. 93, 237-246 (1983; Zbl 0528.20053)], and the results of the reviewer [ibid. 99, 45-50 (1986; Zbl 0583.20052)] on the radical of semigroup algebras satisfying polynomial identities.
semigroup algebras satisfying polynomial identities
scientific article; zbMATH DE number 4067250
scientific article