Indecomposable decomposition and couniserial dimension

A Ghorbani,Z Nazemian,S K Jain

doi:10.1007/s13373-014-0062-6

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. We define and study couniserial dimension for modules. Couniserial dimension is a measure of how far a module deviates from being uniform. Despite their different objectives, it turns out that there are certain common properties between the couniserial dimension and Krull dimension. Among others, each module having such a dimension contains a uniform submodule and has finite uniform dimension. 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, a von Neumann regular ring with countable couniserial dimension is semisimple artinian. If the maximal right quotient ring of a semiprime right 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.

Highlights

In this article we introduce a notion of dimension of a module, to be called couniserial dimension
It is shown in Theorem 4.3 that a module of countable couniserial dimension can be decomposed into indecomposable modules
We show that there exists a prime right Goldie ring R such that c.u.dim(RR) = 2 and Q R does not have couniserial dimension

Summary

Introduction

In this article we introduce a notion of dimension of a module, to be called couniserial dimension. 5, we give some applications of couniserial dimension It is shown in Proposition 5.2 that a module M with finite length is semisimple if and only if for every submodule N of M the right R-module ⊕i∞=1 M/N has couniserial dimension. As a consequence a commutative noetherian ring R is semisimple if and only if for every finite length module M the module ⊕i∞=1 M has couniserial dimension It is shown in Proposition 5.4 that if P is an anti-coHopfian projective right R-module and ⊕i∞=1 E(P) has couniserial dimension, P is injective. As another application we show that all right (left) R-module have couniserial dimension if and only if R is semisimple artinian (see Theorem 5.8). Several examples are included in the paper that demonstrates as to why the conditions imposed are necessary

Definitions and notations

Basic and preliminary results

Main results

Some applications

Summary