Some remarks on generalizations of classical prime submodules

Document Type : Research Paper


Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences, Semnan University, Semnan, Iran.



Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Suppose that $\phi:S(M)\rightarrow S(M)\cup \lbrace\emptyset\rbrace$ be a function where $S(M)$ is the set of all submodules of $M$. A proper submodule $N$ of $M$ is called an $(n-1, n)$-$\phi$-classical prime submodule, if whenever $r_{1},\ldots,r_{n-1}\in R$ and $m\in M$ with $r_{1}\ldots r_{n-1}m\in N\setminus\phi(N)$, then $r_{1}\ldots r_{i-1}r_{i+1}\ldots r_{n-1}m\in N$, for some $i\in\lbrace 1,\ldots, n-1\rbrace$ $(n\geqslant 3)$.
In this work, $(n-1, n)$-$\phi$-classical prime submodules are studied and some results are established.


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