Every module is an inverse limit of injective modules

TL;DR

This paper demonstrates that any left module over a ring can be expressed as an inverse limit of injective modules, providing a new perspective on module decomposition and injective resolutions.

Contribution

It introduces a novel representation of modules as inverse limits of injective modules, extending known results to broader classes of rings.

Findings

Modules can be expressed as inverse limits of injective modules.

Over left Noetherian rings, modules can be represented as inverse limits of surjective homomorphisms of injectives.

The paper raises questions about further implications of these representations.

Abstract

It is shown that any left module A over a ring R can be written as the intersection of a downward directed system of injective submodules of an injective module; equivalently, as an inverse limit of one-to-one homomorphisms of injectives. If R is left Noetherian, A can also be written as the inverse limit of a system of surjective homomorphisms of injectives. Some questions are raised.