Associated primes and primal decomposition of modules over commutative rings

• Autorzy: Ahmad Khojali , Reza Naghipour
• Języki publikacji: 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.

Kategorie tematyczne

• 16D10: General module theory
• 13C13: Other special types

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2009
• Tom: 114
• Numer: 2
• Strony: 191-202

Daty

wydano

2009

Twórcy

autor

Ahmad Khojali

• Department of Mathematics, University of Tabriz, Tabriz 51666-16471, Iran

autor

Reza Naghipour

• 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

Identyfikatory

DOI

10.4064/cm114-2-3