# CHARACTERIZATION OF PRIME SUBMODULES OF A FREE MODULE OF FINITE RANK OVER A VALUATION DOMAIN

• Mirzaei, Fatemeh ;
• Nekooei, Reza
• Received : 2015.10.01
• Published : 2017.01.01
• 56 18

#### 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 $Pr{\ddot{u}}fer$ 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;$Pr{\ddot{u}}fer$ domains;prime submodules

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