论文标题
由模块的Mod-Nihililator代表的基本理想
Essential ideals represented by mod-annihilators of modules
论文作者
论文摘要
令$ r $为具有团结的可交换戒指，$ m $是一个有限的阿贝里安集团（视为$ \ mathbb {z} $ - 模块）。本文的主要目的是研究$ m $的mod-nihilitators的属性。对于$ x \在m $中，我们研究理想$ [x：m] = \ {r \ in r | $ r $的rm \ subseteq rx \} $对应于$ m $的mod-nihihilator。我们调查$ [x：m] $是$ r $的必要理想时。我们证明，由mod-nihililator代表的基本理想的任意交集是必不可少的理想。我们观察到$ [x：m] $在且仅当$ r $是非sinkular而$ r/[x：m] $的激进分子为零时才是配件。此外，如果$ m $的必需socle是非零的，那么我们表明$ [x：m] $是最大理想和$ [x：m]^2 = [x：m] $的交集。最后，我们讨论了$ r $的基本理想的对应关系以及$ m $ $ r $实现的an灭图的顶点。
Let $R$ be a commutative ring with unity, $M$ be a unitary $R$-module and $G$ a finite abelian group (viewed as a $\mathbb{Z}$-module). The main objective of this paper is to study properties of mod-annihilators of $M$. For $x \in M$, we study the ideals $[x : M] =\{r\in R | rM\subseteq Rx\}$ of $R$ corresponding to mod-annihilator of $M$. We investigate that when $[x : M]$ is an essential ideal of $R$. We prove that arbitrary intersection of essential ideals represented by mod-annihilators is an essential ideal. We observe that $[x : M]$ is injective if and only if $R$ is non-singular and the radical of $R/[x : M]$ is zero. Moreover, if essential socle of $M$ is non-zero, then we show that $[x : M]$ is the intersection of maximal ideals and $[x : M]^2 = [x : M]$. Finally, we discuss the correspondence of essential ideals of $R$ and vertices of the annihilating graphs realized by $M$ over $R$.
