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Associated primes and primal decomposition of modules over commutative rings

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Khojali A., Naghipour R.

Colloquium Mathematicum

2009

tom 114

nr 2

191-202

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.

Ograniczanie wyników

1 Colloquium Mathematicum

1 Khojali A.

1 Naghipour R.

1 2009

Wyniki wyszukiwania

Associated primes and primal decomposition of modules over commutative rings