Associated primes and primal decomposition of modules over commutative rings

Ahmad Khojali¹ ; Reza Naghipour²

¹ Department of Mathematics University of Tabriz Tabriz 51666-16471, Iran
² Department of Mathematics University of Tabriz Tabriz 51666-16471, Iran and School of Mathematics Institute for Studies in Theoretical Physics and Mathematics (IPM) P.O. Box 19395-5746, Tehran, Iran

Colloquium Mathematicum, Tome 114 (2009) no. 2, pp. 191-202

Résumé

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.

Détail

DOI : 10.4064/cm114-2-3

Keywords: commutative ring r module paper establish efficient decomposition proper submodule intersection primal submodules prove existence canonical primal decomposition bigcap mathfrak mathfrak where intersection taken isolated components mathfrak primal submodules having distinct incomparable adjoint prime ideals mathfrak using decomposition prove mathfrak mathop supp submodule nbsp intersection mathfrak primal submodules only elements setminus mathfrak prime nbsp shown arithmetical r module only every primal submodule irreducible finally determine conditions canonical primal decomposition irredundant residually maximal unique decomposition irredundant intersection isolated components