Document Type : Research Paper

Authors

• Mohammad Hosein Moslemi Koopaei ¹
• Masoud Zolfaghari ²

¹ Department of Mathematics, Roudehen Branch, Islamic Azad University, Roudehen, Iran

² Department of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran

Abstract

Let $R$ be a commutative ring with identity,  $M$ be an unitary $R$-module, let $\mathcal{S}(M)$ be the set of all submodules of $M$ and $\phi : \mathcal{S}(M)\rightarrow \mathcal{S}(M)\cup \lbrace\emptyset\rbrace$ be a function. A proper submodule $N$ of $M$ is called $\phi$-pimary submodule if $rx\in N \setminus \phi(N)$ where $r\in R$ and $x\in M$, implies that $x\in N$ or $r\in \sqrt{(N:M)}$. In this work, $\phi$-primary submodules are studied, and some results are obtained.

Keywords

• ϕ-prime ideal
• ϕ-primary ideal
• ϕ-prime submodule
• ϕ-primary submodule