Noetherian Semigroup Algebras

Abstract

It is shown that a semigroup S is finitely generated whenever the semigroup algebra K[S] is right Noetherian and has finite Gelfand–Kirillov dimension or S is a Malcev nilpotent semigroup. If, furthermore, S is a submonoid of a finitely generated nilpotent-by-finite group G, then K[S] is right Noetherian if and only if K[S] is left Noetherian, or equivalently S satisfies the ascending chain condition on right (left) ideals. The latter condition is completely described in terms of the structure of S: in case G is a nilpotent group the quotient group H of S contains a normal subgroup F such that H/F is abelian-by-finite and F⊆S. Finally, also prime Goldie contracted semigroup algebras are described.