A note on the construction of finitely injective modules

TL;DR

This paper introduces a new method to construct non-trivial finitely injective modules and characterizes left noetherian rings through the triviality of such modules, answering an open question.

Contribution

It provides a novel construction technique for finitely injective modules and establishes a characterization of left noetherian rings based on these modules.

Findings

Finitely injective modules can be non-trivial, not just direct sums of injectives.

A ring is left noetherian iff all finitely injective modules are trivial.

Answers an open question by Salce regarding finitely injective modules.

Abstract

We develop a technique to construct finitely injective modules which are non trivial, in the sense that they are not direct sums of injective modules. As a consequence, we prove that a ring $R$ is left noetherian if and only if each finitely injective left $R$-module is trivial, thus answering an open question posed by Salce.