Document Type : Original Manuscript

Author

Department of Mathematics, Shahrekord University, P.O.Box 115, Shahrekord, Iran.

Abstract

Let $R$ be a commutative ring with identity and let $M$ be an $R$-module. A proper submodule $P$ of $M$ is called strongly prime submodule if $(P + Rx : M)y P$ for $x, y M$, implies that $x P$ or $y P$. In this paper, we study more properties of strongly prime submodules. It is shown that a finitely generated $R$-module $M$ is Artinian if and only if $M$ is Noetherian and every strongly prime submodule of $M$ is maximal. We also study the strongly dimension of a module
which is defined to be the length of a longest chain of strongly prime submodules.

Keywords