Hereditary semigroup algebras
Let \(K\) be a ring with an identity and \(S\) a monoid. The paper is concerned with the homological dimension of the semigroup ring \(K[S]\) and with conditions for \(K[S]\) to be hereditary, for certain special classes of semigroups \(S\). First, if \(S\) has a finite ideal chain with factors that are non-null Rees matrix semigroups over certain groups, then an upper bound on \(\text{l.gl.dim}(K[S])\) is obtained in terms of the dimension of \(K\), dimensions of group rings \(K[G_j]\) for maximal subgroups \(G_j\) of \(S\), and of some structural invariants of \(S\). This sharpens a result of \textit{J. Kuzmanovich} and \textit{M. L. Teply} [Commun. Algebra 25, 2817-2837 (1997; Zbl 0879.16015)]. A description of hereditary rings \(K[S]\), where \(K\) is a field and \(S\) is a non-null Rees matrix semigroup with the identity adjoined, is derived. Next, for any Malcev nilpotent semigroup \(S\) and any field \(K\), it is shown that the contracted semigroup ring \(K_0[S]\) is a hereditary Noetherian prime ring if and only if \(K_0[S]\) is isomorphic to a full matrix ring over \(K\), \(K[X]\) or \(K[X,X^{-1}]\), and a description of the underlying \(S\) follows. Finally, for any cancellative monoid \(S\), Noetherian hereditary rings \(K[S]\) are fully characterized. Due to an error in typesetting, the reference numbers and cross-references were omitted. The full corrected text has been published in the Erratum.
scientific article; zbMATH DE number 1502163
scientific article