Regular divisors of a submodule

Document Type : Research Paper

Authors

1 Department of Mathematics, Pondicherry University, Puducherry, India.

2 Department of Mathematics, Pondicherry University, Pondicherry, India.

3 Central Institute of Petrochemicals Engineering & Technology, Chennai, Tamil Nadu, India.

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.

Keywords

References

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[5] K. R. Thulasi, T. Duraivel and S. Mangayarcarassy, Generalized prime ideal factorization of submodules, J. Algebra Relat. Top., 9 No. 2 (2021) 121-129.
Algebraic Structures and Their Applications
Volume 10, Issue 2
August 2023
Pages 51-64

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