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Abstract

Let $R$ be a commutative ring and let $M$ be an $R$-module. The aim of this paper is to establish an efficient decomposition of a proper submodule $N$ of $M$ as an intersection of primal submodules. We prove the existence of a canonical primal decomposition, $N=\bigcap_{\mathfrak{p}} N_ {(\mathfrak{p})}$, where the intersection is taken over the isolated components $N_{(\mathfrak{p})}$ of $N$ that are primal submodules having distinct and incomparable adjoint prime ideals $\mathfrak{p}$. Using this decomposition, we prove that for $\mathfrak{p}\in \mathop{\rm Supp}(M//N)$, the submodule $N$ is an intersection of $\mathfrak{p}$-primal submodules if and only if the elements of $R\setminus \mathfrak{p}$ are prime to $N$. Also, it is shown that $M$ is an arithmetical $R$-module if and only if every primal submodule of $M$ is irreducible. Finally, we determine conditions for the canonical primal decomposition to be irredundant or residually maximal, and for the unique decomposition of $N$ as an irredundant intersection of isolated components.