Associated primes and primal decomposition of modules over commutative rings

Ahmad Khojali, and Reza Naghipour. "Associated primes and primal decomposition of modules over commutative rings." Colloquium Mathematicae 114.2 (2009): 191-202. <http://eudml.org/doc/286317>.

@article{AhmadKhojali2009,
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, $N = ⋂_\{\} N_\{()\}$, where the intersection is taken over the isolated components $N_\{()\}$ 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.},
author = {Ahmad Khojali, Reza Naghipour},
journal = {Colloquium Mathematicae},
keywords = {Associated prime ideal; Krull associated prime ideal; weakly associated prime ideal; Zariski-Samuel associated prime ideal; primal submodule; primal decomposition; arithmetical module; irreducible submodule; isolated component; etc.},
language = {eng},
number = {2},
pages = {191-202},
title = {Associated primes and primal decomposition of modules over commutative rings},
url = {http://eudml.org/doc/286317},
volume = {114},
year = {2009},
}

TY - JOUR
AU - Ahmad Khojali
AU - Reza Naghipour
TI - Associated primes and primal decomposition of modules over commutative rings
JO - Colloquium Mathematicae
PY - 2009
VL - 114
IS - 2
SP - 191
EP - 202
AB - 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, $N = ⋂_{} N_{()}$, where the intersection is taken over the isolated components $N_{()}$ 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.
LA - eng
KW - Associated prime ideal; Krull associated prime ideal; weakly associated prime ideal; Zariski-Samuel associated prime ideal; primal submodule; primal decomposition; arithmetical module; irreducible submodule; isolated component; etc.
UR - http://eudml.org/doc/286317
ER -