On stable-projective and injective-costable decompositions of modules

TL;DR

This paper characterizes when modules over certain rings can be decomposed into stable and projective or injective and costable parts, revealing deep categorical and structural properties of the rings.

Contribution

It establishes new equivalences and criteria for module decompositions over left hereditary rings, including functorial and categorical aspects, and generalizes classical module theorems.

Findings

Decomposition of modules into stable and projective parts characterizes left perfect and right coherent rings.

Injective-costable decompositions are unique iff the ring is left hereditary and Noetherian.

Krull-Schmidt property in module categories implies the ring is left Artinian.

Abstract

It is proved that, for a left hereditary ring, an arbitrary left module has a representation in the form of the direct sum of a stable left module and indecomposable projective left modules (if and only if an arbitrary left module has a representation in the form of the direct sum of a stable left module and a projective left module) if and only if the ring is left perfect and right coherent. In that case, the above-mentioned representations are unique up to isomorphism; the latter representation is also functorial. The essential ingredient in the proofs of the above-mentioned statements is a certain purely categorical result. These statements, in particular, imply that, for any principal ideal domain that is not a field, the fundamental theorem on finitely generated modules over it can not be generalized to the case of all modules. Moreover, with the aid of the above-mentioned categorical approach, we give a new proof of the Zheng-Xu He's result asserting that any module of a ring has a unique up to isomorphism injective-costable decomposition if and only if the ring is left hereditary and left Noetherian. The above-mentioned statements, in particular, imply that if the category of left modules over a left hereditary ring is Krull-Schmidt, then the ring is left Artinian. Yet another criterion for a ring to be left hereditary, left perfect and right coherent (resp. left hereditary left Noetherian) found in the paper requires that the pair $(Stable$ $modules$, $Projective$ $modules)$ (resp. $(Injective$ $Modules, Costable$ $ Modules)$) of module classes be a pre-torsion theory. This implies that the pair $(Stable$ $modules$, $Projective$ $modules)$ is a torsion theory if and only if the ring is left hereditary and the injective envelope of the ring, viewed as a left module over itself, is projective.