A short Note on prime submodules

Document Type : Research Paper


Department of mathematics, Faculty of sciences, University of Mohaghegh Ardabili, Ardabil, Iran.


Let $R$ be a commutative ring with identity and $M$ be a unital $R$-module. A proper submodule $N$ of $M$ with $N:_RM=\frak p$ is said to be prime or $\frak p$-prime ($\frak p$ a prime ideal of $R$) if $rx\in N$ for $r\in R$ and $x\in M$ implies that either $x\in N$ or $r\in \frak p$. In this paper we study a new equivalent conditions for a minimal prime submodules of an $R$-module to be a finite set, whenever $R$ is a Noetherian ring. Also we introduce the concept of arithmetic rank of a submodule of a Noetherian module and we give an upper bound for it.


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