Co-Kasch Modules
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866929739888852992 |
|---|---|
| author | Alizade, Rafail Büyükaşık, Engin Durgun, Yılmaz |
| author_facet | Alizade, Rafail Büyükaşık, Engin Durgun, Yılmaz |
| contents | 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 $σ[M]$ 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 $σ[P]$. 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 Prüfer 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04059 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Co-Kasch Modules Alizade, Rafail Büyükaşık, Engin Durgun, Yılmaz Rings and Algebras 16D10, 16D90, 16P20, 16P40 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 $σ[M]$ 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 $σ[P]$. 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 Prüfer 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. |
| title | Co-Kasch Modules |
| topic | Rings and Algebras 16D10, 16D90, 16P20, 16P40 |
| url | https://arxiv.org/abs/2409.04059 |