FP-injectivity of factors of injective modules

TL;DR

This paper characterizes rings based on the FP-injectivity of factors of injective modules, linking properties like semiheredity, reducedness, and arithmetical structure to FP-injectivity conditions.

Contribution

It provides new characterizations of rings such as semihereditary, reduced, and arithmetical rings through FP-injectivity of modules and their factors, extending understanding of module and ring theory.

Findings

A ring is left semihereditary iff each homomorphic image of its injective hull is FP-injective.

A commutative ring is reduced and arithmetical iff certain quotients of FP-injective modules are FP-injective.

In chain rings, conditions for FP-injectivity of factors relate to the completeness of the quotient ring in the f.c. topology.

Abstract

It is shown that a ring is left semihereditary if and only each homomorphic image of its injective hull as left module is FP-injective. It is also proven that a commutative ring R is reduced and arithmetical if and only if E/U if FP-injective for any FP-injective R-module E and for any submodule U of finite Goldie dimension. A characterization of commutative rings for which each module of finite Goldie dimension is of injective dimension at most one is given. Let R be a chain ring and Z its subset of zerodivisors. It is proven that E/U is FP-injective for each FP-injective R-module E and each pure polyserial submodule U of E if R/I is complete in its f.c. topology for each ideal I whose the top prime ideal is Z. The converse holds if each indecomposable injective module whose the bottom prime ideal is Z contains a pure uniserial submodule. For some chain ring R we show that E/U is FP-injective for any FP-injective module E and any its submodule U of finite Goldie dimension, even if R is not coherent. It follows that any Archimedean chain ring is either coherent or maximal if and only if each factor of any injective module of finite Goldie dimension modulo a pure submodule is injective.