Module decompositions using pairwise comaximal ideals

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.