Ideals as Generalized Prime Ideal Factorization of Submodules

Ideals as Generalized Prime Ideal Factorization of Submodules

K. R. Thulasi, T. Duraivel, S. Mangayarcarassy

Abstract

For a submodule $N$ of an $R$ -module $M$ , a unique product of prime ideals in $R$ is assigned, which is called the generalized prime ideal factorization of $N$ in $M$ , and denoted as ${\mathcal{P}}_M(N)$ . But for a product of prime ideals ${{{\mathfrak{p}}_1} \cdots {{\mathfrak{p}}_{n}}}$ in $R$ and an $R$ -module $M$ , there may not exist a submodule $N$ in $M$ with ${\mathcal{P}}_{M}(N) = {{{\mathfrak{p}}_1} \cdots {{\mathfrak{p}}_{n}}}$ . In this article, for an arbitrary product of prime ideals ${{{\mathfrak{p}}_1} \cdots {{\mathfrak{p}}_{n}}}$ and a module $M$ , we find conditions for the existence of submodules in $M$ having ${{{\mathfrak{p}}_1} \cdots {{\mathfrak{p}}_{n}}}$ as their generalized prime ideal factorization

Keywords: prime submodule; prime filtration; Noetherian ring; prime ideal factorization; regular prime extension filtration

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Note di Matematica

Ideals as Generalized Prime Ideal Factorization of Submodules

Abstract