Noetherian semigroup algebras
Let \(S\) be a semigroup and let \(K\) be a field. The authors are interested in conditions under which the semigroup algebra \(K[S]\) is right or left Noetherian. In particular, they investigate cases when these two concepts coincide, when Noetherian property of \(K[S]\) implies \(S\) is finitely generated, and when \(S\) satisfies the ascending chain conditions on one-sided ideals. Main results sound as follows. Suppose that \(S\) is a submonoid of a finitely generated nilpotent-by-finite group; then the following three conditions are equivalent: \(K[S]\) is right Noetherian, \(K[S]\) is left Noetherian, \(S\) satisfies the ascending chain conditions on one-sided ideals (Corollary 1.6). Suppose that either \(S\) is a submonoid of a finitely generated nilpotent-by-finite group, or \(K[S]\) has finite Gelfand-Kirillov dimension, or else \(S\) is Malcev nilpotent; then \(S\) is finitely generated (Theorem 2.2). A description of prime Goldie semigroup algebras is also given, a generalization of Zelmanov's well known results on prime PI-semigroup algebras.
finitely generated nilpotent-by-finite groups
scientific article; zbMATH DE number 1348171
scientific article