论文标题
弱变形模块
Weakly-morphic modules
论文作者
论文摘要
令$ r $为可交换戒指，$ m $ a $ r $ - 模块和$φ_A$是右乘以$ a \ in r $ $ a \ in r $ $ a \ in r $ in $ m $的内态性。我们说，如果$ m/φ_a（m）\ cong \ ker（φ_a）$是$ r $ -modules，则$ m $是{\ it弱变形}。我们研究这些模块，并使用它们来表征环$ r/\ text {ann} _r（m）$，其中$ \ text {ann} _r（m）$是$ m $的正确歼灭者。当且仅当$ r/\ text {ann} _r（m）$的每个元素作为$ m $的内态元素时，且仅当$ r/\ text {ann} _r（m）$的每个元素都是$ m $的$ r/\ text {ann} _r（m）$时，内核直接或图像导向模块$ m $。如果$ m $是一个积分域$ r $上的弱变形模块，则$ m $在且仅当$ r/\ text {ann} _r（m）$是一个字段时，只有当时是无扭力的。有限生成的$ \ bbb z $ - 模块在有限的情况下才是弱态的；并且仅当它是弱变形的时，它是形态的，并且其每个主要组件的形式为$（\ bbb z_ {p^k}）^n $，对于某些非阴性整数$ n $和$ k $。
Let $R$ be a commutative ring, $M$ an $R$-module and $φ_a$ be the endomorphism of $M$ given by right multiplication by $a\in R$. We say that $M$ is {\it weakly-morphic} if $M/φ_a(M)\cong \ker(φ_a)$ as $R$-modules for every $a$. We study these modules and use them to characterise the rings $R/\text{Ann}_R(M)$, where $\text{Ann}_R(M)$ is the right annihilator of $M$. A kernel-direct or image-direct module $M$ is weakly-morphic if and only if each element of $R/\text{Ann}_R(M)$ is regular as an endomorphism element of $M$. If $M$ is a weakly-morphic module over an integral domain $R$, then $M$ is torsion-free if and only if it is divisible if and only if $R/\text{Ann}_R(M)$ is a field. A finitely generated $\Bbb Z$-module is weakly-morphic if and only if it is finite; and it is morphic if and only if it is weakly-morphic and each of its primary components is of the form $(\Bbb Z_{p^k})^n$ for some non-negative integers $n$ and $k$.
