On special submodule of modules

Document Type : Research Paper


1 iranian

2 Buali sina University


‎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 that an $R$-module $M$ admits a‎
‎powerful submodule if and only if it admits a strongly primary‎
‎submodule‎. ‎Finally we study finitely generated torsion free‎
‎modules over domain each of whose prime submodules are strongly‎


Main Subjects