Localization of injective modules over arithmetical rings

TL;DR

This paper investigates conditions under which localizations of injective modules over arithmetical rings remain injective, focusing on properties like coherence, semicoherence, and finite Goldie dimension.

Contribution

It establishes new criteria ensuring localizations of injective modules are injective over arithmetical rings with specific properties.

Findings

Localizations of injective modules are injective if R_P is coherent or not semicoherent.

Finitely injective modules localize to finitely injective modules under certain conditions.

Over Pr"ufer domains of finite character, localizations of injective modules are always injective.

Abstract

It is proved that localizations of injective $R$-modules of finite Goldie dimension are injective if $R$ is an arithmetical ring satisfying the following condition: for every maximal ideal $P$, $R_P$ is either coherent or not semicoherent. If, in addition, each finitely generated $R$-module has finite Goldie dimension, then localizations of finitely injective $R$-modules are finitely injective too. Moreover, if $R$ is a Pr\"ufer domain of finite character, localizations of injective $R$-modules are injective.