An introduction to homological algebra: by D. G. Northcott. 282 pages, diagrams, 6 × 9 in. New York, Cambridge University Press, 1960. Price, $8.00

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.