Regular divisors of a submodule

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Duraivel T.; Thulasi K. R.; Premkumar K.

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Journal Paper

Paper Information

Journal: Algebraic structures and their applications Year:2023 | Volume:10 | Issue:2 Page(s): 51-64

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Information Journal Paper

Title

Regular divisors of a submodule

Author(s)

Duraivel T. | Thulasi K. R. | Premkumar K. | Issue Writer Certificate

Keywords

Maximal independent subset

Partially ordered set

Prime ideal factorization

Regular divisors

Regular prime extension filtration

Abstract

In this article, we extend the concept of divisors to ideals of Noetherian rings, more generally, to submodules of finitely generated modules over Noetherian rings. For a submodule $N$ of a finitely generated module $M$ over a Noetherian ring, we say a submodule $K$ of $M$ is a regular divisor of $N$ in $M$ if $K$ occurs in a Regular prime extension filtration of $M$ over $N$. We show that a submodule $N$ of $M$ has only a finite number of Regular divisors in $M$. We also show that an ideal $\mathfrak b$ is a regular divisor of a non-zero ideal $\mathfrak a$ in a Dedekind domain $R$ if and only if $\mathfrak b$ contains $\mathfrak a$. We characterize Regular divisors using some ordered sequences of prime ideals and study their various properties. Lastly, we formulate a method to compute the number of Regular divisors of a submodule by solving a combinatorics problem.

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