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 $(p + rx : m)ysubseteq p$ for $x, yin m$, implies that $xin p$ or $yin p$. 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 st...

let r be a g-graded ring and m be a g-graded r-module. in this article, we introduce the concept of graded weakly classical prime submodules and give some properties of such submodules.

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

For any field F , we give an example of a prime finitely generated F -algebra of GK dimension 2 for which GK dimension is not finitely partitive. For each m ∈ {0, 1, 2, 3, . . .}∪{∞}, we give an example of a prime finitely generated F -algebra having GK dimension 2 which has classical Krull dimension m; the algebra of GK dimension 2 which has infinite classical Krull dimension has the additiona...

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 $(P + Rx : M)ysubseteq P$ for $x, yin M$, implies that $xin P$ or $yin P$. 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 st...

let $n$ be a submodule of a module $m$ and a minimal primary decomposition of $n$ is known‎. ‎a formula to compute baer's lower nilradical of $n$ is given‎. ‎the relations between classical prime submodules and their nilradicals are investigated‎. ‎some situations in which semiprime submodules can be written as finite intersection of classical prime submodule are stated‎.

‎let $g$ be a group with identity $e.$ let $r$ be a $g$-graded‎ ‎commutative ring and $m$ a graded $r$-module‎. ‎in this paper‎, ‎we‎ ‎introduce several results concerning graded classical prime‎ ‎submodules‎. ‎for example‎, ‎we give a characterization of graded‎ ‎classical prime submodules‎. ‎also‎, ‎the relations between graded‎ ‎classical prime and graded prime submodules of $m$ are studied‎.‎