Additive Rank Functions and Chain Conditions

Abstract

Let R be a ring with identity and let mod-R denote the category of finitely generated unitary right R-modules. By an additive rank function on mod-R we mean a mapping ρ from mod-R to the set ℕ of non-negative integers such that, for every exact sequence 0 → L → M → H → 0 in mod-R, we have ρ(M)=ρ(L)+ρ(H). In this note we exhibit a relationship between additive rank functions on mod-R and ascending chain conditions of a certain type. Specifically, we prove that if R is a right Noetherian ring, ρ is an additive rank function on mod-R and a right R-module M=Π _i∈I M _i is a direct product of finitely generated submodules M _i (i ∈ I) such that ρ(L)≠0 for every non-zero submodule L of M _i for each i ∈ I then M satisfies the ascending chain condition on n-generated submodules for every positive integer n. In particular this implies that, for a right Noetherian ring R which is an order in a right Artinian ring, every torsionless right R-module satisfies the ascending chain condition on n-generated submodules for every positive integer n. It is also proved that if R is a right Noetherian ring which satisfies the descending chain condition on right annihilators then every torsionless right R-module M such that every countably generated submodule is contained in a direct sum of finitely generated submodules of M satisfies the ascending chain condition on n-generated submodules for every positive integer n.