‎let $i$ denote an ideal of a noetherian ring $r$‎. ‎the purpose of‎ ‎this article is to introduce the concepts of quintasymptotic‎ ‎sequences over $i$ and quintasymptotic cograde of $i$‎, ‎and to show that they play a role analogous to quintessential sequences‎ ‎over $i$ and quintessential cograde of $i$‎. ‎we show that‎, ‎if $r$ is‎ ‎local‎, ‎then the quintasymptotic cograde of $i$ is una...

‎Let $I$ denote an ideal of a Noetherian ring $R$‎. ‎The purpose of‎ ‎this article is to introduce the concepts of quintasymptotic‎ ‎sequences over $I$ and quintasymptotic cograde of $I$‎, ‎and to show that they play a role analogous to quintessential sequences‎ ‎over $I$ and quintessential cograde of $I$‎. ‎We show that‎, ‎if $R$ is‎ ‎local‎, ‎then the quintasymptotic cograde of $I$ is una...

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 $r$ be a domain with quotiont field $k$‎, ‎and‎ ‎let $n$ be a submodule of an $r$-module $m$‎. ‎we say that $n$ is‎ ‎powerful (strongly primary) if $x,yin k$ and‎ ‎$xymsubseteq n$‎, ‎then $xin r$ or $yin r$ ($xmsubseteq n$‎ ‎or $y^nmsubseteq n$ for some $ngeq1$)‎. ‎we show that a submodule‎ ‎with either of these properties is comparable to every prime‎ ‎submodule of $m$‎, ‎also we show tha...

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.