Co-Kasch Modules

Rafail Alizade, Engin Büyükaşık
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Abstract

In this paper we study the modules $M$ every simple subfactors of which is a homomorphic image of $M$ and call them co-Kasch modules. These modules are dual to Kasch modules $M$ every simple subfactors of which can be embedded in $M$. We show that a module is co-Kasch if and only if every simple module in $\sigma[M]$ is a homomorphic image of $M$. In particular, a projective right module $P$ is co-Kasch if and only if $P$ is a generator for $\sigma[P]$. If $R$ is right max and right $H$-ring, then every right $R$-module is co-Kasch; and the converse is true for the rings whose simple right modules have locally artinian injective hulls. For a right artinian ring $R$, we prove that: (1) every finitely generated right $R$-module is co-Kasch if and only if every right $R$-module is a co-Kasch module if and only if $R$ is a right $H$-ring; and (2) every finitely generated projective right $R$-module is co-Kasch if and only if the Cartan matrix of $R$ is a diagonal matrix. For a Pr\"ufer domain $R$, we prove that, every nonzero ideal of $R$ is co-Kasch if and only if $R$ is Dedekind. The structure of $\mathbb{Z}$-modules that are co-Kasch is completely characterized.
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共用卡什模块
在本文中,我们研究了每一个简单子因子都是 $M$ 的同构像的模块 $M$,并称它们为共卡什模块。我们证明,当且仅当$\sigma[M]$ 中的每个简单模块都是$M$ 的同构像时,模块才是共卡什模块。尤其是,当且仅当 $P$ 是 $\sigma[P]$ 的生成器时,一个投影右模块 $P$ 是共卡什模块。如果$R$是右最大和右$H$环,那么每个右$R$模块都是共卡斯;反之亦然,对于其简单右模块具有局部自洽注入环的环来说也是如此。对于右artinian 环 $R$,我们证明(1)当且仅当 $R$ 是一个右 $H$ 环时,每一个有限生成的右 $R$ 模块都是共卡斯模块;(2)当且仅当 $R$ 的 Cartan 矩阵是一个对角矩阵时,每一个有限生成的投影右 $R$ 模块都是共卡斯模块。对于一个 Pr\"ufer 域$R$,我们证明,当且仅当 $R$ 是 Dedekind 时,$R$ 的每一个非零理想都是 co-Kasch。共卡斯的 $\mathbb{Z}$ 模块的结构被完整地描述了。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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