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Colloquium Mathematicum

2009 | 114 | 2 | 191-202
Tytuł artykułu

Associated primes and primal decomposition of modules over commutative rings

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EN
Abstrakty
EN
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, $N = ⋂_{𝔭} N_{(𝔭)}$, where the intersection is taken over the isolated components $N_{(𝔭)}$ of N that are primal submodules having distinct and incomparable adjoint prime ideals 𝔭. Using this decomposition, we prove that for 𝔭 ∈ Supp(M/N), the submodule N is an intersection of 𝔭-primal submodules if and only if the elements of R∖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.
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Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
191-202
Opis fizyczny
Daty
wydano
2009
Twórcy
autor
• Department of Mathematics, University of Tabriz, Tabriz 51666-16471, Iran
autor
• Department of Mathematics, University of Tabriz, Tabriz 51666-16471, Iran
• School of Mathematics, Institute for Studies, in Theoretical Physics and Mathematics (IPM), P.O. Box 19395-5746, Tehran, Iran
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