Indecomposable injective modules of finite Malcev rank over local commutative rings

TL;DR

This paper characterizes when indecomposable injective modules over certain local rings are polyserial, linking this property to the finiteness of ranks over extensions, and explores related invariants and conditions.

Contribution

It provides new criteria for indecomposable injective modules to be polyserial over valuation and chain rings, relating module invariants to ring extension properties.

Findings

Indecomposable injective modules over valuation domains are polyserial iff maximal immediate extensions have finite rank.

Finiteness of Malcev, Fleischer, and dual Goldie invariants for these modules under certain conditions.

Indecomposable injective modules over Krull-dimensional local Noetherian rings have finite Malcev rank.

Abstract

It is proven that each indecomposable injective module over a valuation domain $R$ is polyserial if and only if each maximal immediate extension $\widehat{R}$ of $R$ is of finite rank over the completion $\widetilde{R}$ of $R$ in the $R$-topology. In this case, for each indecomposable injective module $E$, the following invariants are finite and equal: its Malcev rank, its Fleischer rank and its dual Goldie dimension. Similar results are obtained for chain rings satisfying some additional properties. It is also shown that each indecomposable injective module over one Krull-dimensional local Noetherian rings has finite Malcev rank. The preservation of Goldie dimension finiteness by localization is investigated too.