Noetherian Semigroup Rings with Several Objects

Abstract

Whether the following generalized version of Hilbert's Basis Theorem GHBT: C - Mod locally noetherian → C [ X ]- Mod locally noetherian, holds or not depends on the small category X and possibly likewise on the category C - Mod of modules over a ring C with several objects in the sense of B. Mitchell [11]; C [ X ]:= C ⊗Z[ X ] generalizes the concept of semigroup rings. There is only little hope to characterize all small categories X for which GHBT holds because of the special case C = k being a commutative field and X = (G,˙) being a group. Then C [ X ]- Mod is just the category of left modules over the group ring k[G]. It is a rather old unsolved problem to distinguish all groups for which k[G] is left noetherian [12,13]. We consider a class of certain order enriched small categories including all path categories of quivers, free commutative monoids, posets, and others; GHBT holds for such a category and all C iff it fulfils the ascending chain condition on subfunctors of its covariant morphism functors. The classical Hilbert Basis Theorem is a special case of this result.