In this paper, we characterize $w$-Noetherian modules in terms of polynomial modules and $w$-Nagata modules. Then it is shown that for a finite type $w$-module $M$, every $w$-epimorphism of $M$ onto itself is an isomorphism. We also define and study the concepts of $w$-Artinian modules and $w$-simple modules. By using these concepts, it is shown that for a $w$-Artinian module $M$, every $w$-monomorphism of $M$ onto itself is an isomorphism and that for a $w$-simple module $M$, $End_R M$ is a division ring.
展开▼
机译：在本文中，我们用多项式模块和$ w $ -Nagata模块来表征$ w $ -Noetherian模块。然后表明，对于有限类型$ w $ -module $ M $，$ M $自身上的每个$ w $-表象都是同构。我们还定义和研究$ w $ -Artinian模块和$ w $-简单模块的概念。通过使用这些概念，可以看出，对于$ w $ -Artinian模块$ M $，$ M $自身上的每个$ w $-单态都是同构的，对于$ w $-简单模块$ M $， $ End_R M $是一个除法环。
展开▼
