论文标题
分级$ r $ -submodules
Graded $r$-Submodules
论文作者
论文摘要
让$ g $是一个具有身份$ e $的集团和$ r $ a $ g $ g $ g $ raded的戒指，非零统一$ 1 $。在本文中，我们介绍了分级$ r $ submodules和分级的特殊$ r $ $ submodules的概念，这是对分级r-iDeals概念的概括。对于非零$ g $ - $ r $ -module $ m $，适当的分级$ r $ -r $ -subModule $ k $ $ m $ $ m $的分级为$ r $ r $ -submodule（resp。，resp。划分的$ r $ -submodule，n时何时在h（r）$ y in h（r）$ y in h（r）中， $ ann_ {m}（a）= \ {0 \} $（wesp。，$ ann_ {r}（x）= \ {0 \} $），然后$ x \ in k $（wesp。，$ a \ in（k：k：_ _ {r} m）$）。我们研究了分级$ r $ submodules和分级的特殊$ r $ -submodules的各种属性，我们给出了这两个新等级模块的几个插图示例。
Let $G$ be a group with identity $e$ and $R$ a commutative $G$-graded ring with a nonzero unity $1$. In this article, we introduce the concepts of graded $r$-submodules and graded special $r$-submodules, which are generalizations for the notion of graded r-ideals. For a nonzero $G$-graded $R$-module $M$, a proper graded $R$-submodule $K$ of $M$ is said to be graded $r$-submodule (resp., graded special $r$-submodule) if whenever $a\in h(R)$ and $x\in h(M)$ such that $ax\in K$ with $Ann_{M}(a)=\{0\}$ (resp., $Ann_{R}(x)=\{0\}$), then $x\in K$ (resp., $a\in (K:_{R}M)$). We study various properties of graded $r$-submodules and graded special $r$-submodules, and we give several illustration examples of these two new classes of graded modules.
