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, , where the intersection is taken over the isolated components of N that are primal submodules having distinct and incomparable adjoint prime ideals . Using this decomposition, we prove that for ∈ Supp(M/N), the submodule N is an intersection of -primal submodules if and only if the elements of R∖ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm114-2-3,
author = {Ahmad Khojali and Reza Naghipour},
title = {Associated primes and primal decomposition of modules over commutative rings},
journal = {Colloquium Mathematicae},
volume = {116},
year = {2009},
pages = {191-202},
zbl = {1173.13011},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm114-2-3}
}
Ahmad Khojali; Reza Naghipour. Associated primes and primal decomposition of modules over commutative rings. Colloquium Mathematicae, Tome 116 (2009) pp. 191-202. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm114-2-3/