Characterization of prime submodules of a free module of finite rank over a valuation domain

Fatemeh Mirzaei; Reza Nekooei

doi:10.4134/JKMS.j150591

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2017; 54(1): 59-68

Online first article August 25, 2016 Printed January 1, 2017

https://doi.org/10.4134/JKMS.j150591

Characterization of prime submodules of a free module of finite rank over a valuation domain

Fatemeh Mirzaei and Reza Nekooei

Shahid Bahonar University of Kerman, Shahid Bahonar University of Kerman

Abstract

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Abstract

Let $F=R^{(n)}$ be a free $R$-module of finite rank $n\geq 2$. In this paper, we characterize the prime submodules of $F$ with at most $n$ generators when $R$ is a \pr domain. We also introduce the notion of prime matrix and we show that when $R$ is a valuation domain, every finitely generated prime submodule of $F$ with at least $n$ generators is the row space of a prime matrix.

Keywords: Dedekind domains, Pr\"ufer domains, prime submodules

MSC numbers: 13F05, 13C99