Modules whose hereditary pretorsion classes are closed under products

Abstract

A module M is called product closed if every hereditary pretorsion class in σ [M] is closed under products in σ [M]. Every module M which is locally of finite length (every finitely generated submodule of M has finite length) is product closed and every product closed module M is semilocal (M/J (M) is semisimple). Let M ∈ R-Mod be product closed and projective in σ [M]. It is shown that (1) M is semiartinian; (2) if M is finitely generated then M satisfies the DCC on fully invariant submodules; (3) M has finite length if M is finitely generated and every hereditary pretorsion class in σ [M] is M-dominated. If the ring R is commutative it is proven that M is product closed if and only if M is locally of finite length. It was shown by Beachy and Blair [2, Proposition 1.4 and Corollary 3.3] that the following three conditions on a ring R with identity are equivalent: (1) every hereditary pretorsion class in R-Mod is closed under arbitrary (and not just finite) direct products, or equivalently, every left topologizing filter on R is closed under arbitrary (and not just finite) intersections; (2) every left R-module M is finitely annihilated, meaning (0 : M) = (0 : X ) for some finite subset X of M; (3) R is left artinian. In this paper we shall attempt to describe those modules M with the property that every hereditary pretorsion class in the Grothendieck category σ [M] is closed under products in σ [M]. One of our main results says that if M is a finitely generated product closed module such that M is projective in σ [M] and every hereditary pretorsion class in σ [M] is M-dominated (meaning, every hereditary pretorsion class in σ [M] is subgenerated by an M-generated module), then M has finite length. This result extends Beachy and Blair's characterization of left artinian rings. Their proof is based on two results due to Beachy [1, Propositions 1 and 5], but the techniques used by Beachy are not easily generalized in a manner useful for our purposes. We have thus had to develop new methods.

Abelian endoregular modules

“…Note that Θ is surjective because M is quasi-projective. Since M is projective in σ[M ], M/P is projective in σ[M/P ] by [20,Lemma 9]. We have M/P is a prime module, hence End R (M/P ) is a prime ring by [13,Lemma 5.9].…”

“…Since P is fully invariant, every endomorphism ϕ : M → M defines an endomorphism ϕ : M/P → M/P . Then, there exists a ring homomorphism [20,Lemma 9]. We have M/P is a prime module, hence End R (M/P ) is a prime ring by [13,Lemma 5.9].…”

Product closed modules need not be artinian

“…This result provided the motivation for [7] which is devoted to the more general investigation of left R-modules M with the property that every hereditary pretorsion class in σ[M ] is closed under arbitrary products in σ [M ]. A module M with this property is called product closed.…”

“…Beachy and Blair's theorem thus states that the ring R, when considered as left module over itself, is product closed precisely when R is left artinian. Results in [7] show that this characterization fails for a general module M . Whilst a module M with finite length is necessarily…”

“…product closed, the converse need not be true. Indeed, [7,Example 11] exhibits a nonzero product closed module with zero socle and such a module is manifestly nonartinian. This failure in the case of a general M is not surprising, for the ring R, when considered as a module over itself, enjoys a number of special properties: it is finitely generated, projective and is a generator for the category R-Mod.…”