Modules in which every surjective endomorphism has a \(\delta\)-small kernel
In this paper, we introduce the notion of \(\delta\)-Hopfian modules. We give some properties of these modules and provide a~characterization of semisimple rings in terms of \(\delta\)-Hopfian modules by proving that a ring \(R\) is semisimple if and only if every \(R\)-module is \(\delta\)-Hopf...
Saved in:
| Date: | 2019 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2019
|
| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/365 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | In this paper, we introduce the notion of \(\delta\)-Hopfian modules. We give some properties of these modules and provide a~characterization of semisimple rings in terms of \(\delta\)-Hopfian modules by proving that a ring \(R\) is semisimple if and only if every \(R\)-module is \(\delta\)-Hopfian. Also, we show that for a ring \(R\), \(\delta(R)=J(R)\) if and only if for all \(R\)-modules, the conditions \(\delta\)-Hopfian and generalized Hopfian are equivalent. Moreover, we prove that \(\delta\)-Hopfian property is a Morita invariant. Further, the \(\delta\)-Hopficity of modules over truncated polynomial and triangular matrix rings are considered. |
|---|