Associated primes and primal decomposition of modules over commutative rings

Volume 114 / 2009

Ahmad Khojali, Reza Naghipour Colloquium Mathematicum 114 (2009), 191-202 MSC: Primary 16D10, 13C13. DOI: 10.4064/cm114-2-3

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.

Authors

  • Ahmad KhojaliDepartment of Mathematics
    University of Tabriz
    Tabriz 51666-16471, Iran
    e-mail
  • Reza NaghipourDepartment 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
    e-mail

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