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Krull modules and completely integrally closed modules

Mu’amar Musa Nurwigantara

Department of Mathematics, Universitas Gadjah Mada, Sekip Utara, Yogyakarta 55281, Indonesia

E-mail Address: muamar.musa.n@mail.ugm.ac.id

Corresponding author.

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Indah Emilia Wijayanti

Department of Mathematics, Universitas Gadjah Mada, Sekip Utara, Yogyakarta 55281, Indonesia

E-mail Address: ind_wijayanti@ugm.ac.id

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Hidetoshi Marubayashi

Department of Mathematics, Naruto University of Education, Naruto, Tokushima, 72- 8502, Japan

E-mail Address: marubaya@naruto-u.ac.jp

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, and

Sri Wahyuni

Department of Mathematics, Universitas Gadjah Mada, Sekip Utara, Yogyakarta 55281, Indonesia

E-mail Address: swahyuni@ugm.ac.id

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https://doi.org/10.1142/S021949882350038XCited by:4 (Source: Crossref)

Abstract

Let $M$ be a torsion-free module over an integral domain $D$ with quotient field $K$ . We define a concept of completely integrally closed modules in order to study Krull modules. It is shown that a Krull module $M$ is a $v$ -multiplication module if and only if ${(𝔭 M)}_{v}$ is a maximal $v$ -submodule and ${(𝔭^{- 1} 𝔭)}_{v_{1}} = D$ for every minimal prime ideal $𝔭$ of $D$ . If $M$ is a finitely generated Krull module, then $M_{1} = \cap M_{𝔭}$ is a Krull module and $v$ -multiplication module. It is also shown that the following three conditions are equivalent: $M$ is completely integrally closed, $M [x]$ is completely integrally closed, and $M [[x]]$ is completely integrally closed.

Communicated by S. Lopez-Permouth

Keywords:

AMSC: 13B25, 13C05, 13C12, 13E15, 13F05

References

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Vol. 21, No. 01

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History

Received 2 May 2021

Accepted 9 September 2021

Published: 19 November 2021

Information

This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License. which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Krull modules and completely integrally closed modules

Abstract

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