TY - JOUR ID - 1209 TI - A short Note on prime submodules JO - Algebraic Structures and Their Applications JA - AS LA - en SN - 2382-9761 AU - A'zami, Jafar AD - Department of mathematics, Faculty of sciences, University of Mohaghegh Ardabili, Ardabil, Iran. Y1 - 2018 PY - 2018 VL - 5 IS - 1 SP - 41 EP - 49 KW - arithmetic rank of a submodule KW - associated primes KW - height of a prime submodule KW - minimal prime submodule KW - prime submodule DO - 10.22034/as.2018.1209 N2 - 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. UR - https://as.yazd.ac.ir/article_1209.html L1 - https://as.yazd.ac.ir/article_1209_0c73ba791d9609f1a9176263ae0fbd5b.pdf ER -