Module decompositions using pairwise comaximal ideals

TL;DR

This paper establishes a module decomposition theorem using pairwise comaximal ideals, generalizing torsion decompositions in abelian groups, applicable to various classes of rings including semilocal and perfect rings.

Contribution

It introduces a new module decomposition framework based on pairwise comaximal ideals, extending classical torsion theory to broader ring classes.

Findings

Decomposition applies to semilocal and perfect rings.

Generalizes torsion abelian group decompositions.

Develops a related torsion theory for modules.

Abstract

In this paper we show that for a given set of pairwise comaximal ideals $\{X_i\}_{i\in I}$ in a ring $R$ with unity and any right $R$-module $M$ with generating set $Y$ and $C(X_i)=\sum\limits_{k\in\mathbb{N}}\underline{\ell}_M(X_i^{k})$, $M=\oplus_{i\in I}C(X_i)$ if and only if for every $y\in Y$ there exists a nonempty finite subset $J\subseteq I$ and positive integers $k_j$ such that $\bigcap\limits_{j\in J}X_i^{k_j}\subseteq\underline{r}_R(yR)$. We investigate this decomposition for a general class of modules. Our main theorem can be applied to a large class of rings including semilocal rings $R$ with the Jacobson radical of $R$ equal to the prime radical of $R$, left (or right) perfect rings, piecewise prime rings, and rings with ACC on ideals and satisfying the right AR property on ideals. This decomposition generalizes the decomposition of a torsion abelian group into a direct sum of its p-components. We also develop a torsion theory associated with sets of pairwise comaximal ideals.