Indecomposable Decomposition and couniserial dimension

TL;DR

This paper introduces the couniserial dimension, a new ordinal-valued invariant for rings and modules, exploring its properties, relation to other dimensions, and implications for module decomposition and ring structure.

Contribution

It defines the couniserial dimension, analyzes its properties, and establishes its connections to indecomposable decomposition and classical ring-theoretic conditions.

Findings

Modules with countable couniserial dimension have indecomposable decomposition.

Von Neumann regular rings with countable couniserial dimension are semisimple artinian.

Rings where all modules have couniserial dimension are semisimple artinian.

Abstract

Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. In this paper we define one of the dimensions called couniserial dimension that measures how close a ring or module is to being uniform. Despite their different objectives, it turns out that there are certain common properties between the couniserial dimension and Krull dimension like each module having such a dimension contains a uniform submodule and has finite uniform dimension, among others. Like all dimensions, this is an ordinal valued invariant. Every module of finite length has couniserial dimension and its value lies between the uniform dimension and the length of the module. Modules with countable couniserial dimension are shown to possess indecomposable decomposition. In particular, von Neumann regular ring with countable couniserial dimension is semisimple artinian. If the maximal right quotient ring of a non-singular ring R has a couniserial dimension as an R-module, then R is a semiprime right Goldie ring. As one of the applications, it follows that all right R-modules have couniserial dimension if and only if R is a semisimple artinian ring.