On a variation of \(\oplus\)-supplemented modules
Let \(R\) be a ring and \(M\) be an \(R\)-module. \(M\) is called \(\oplus_{ss}\)-supplemented if every submodule of \(M\) has a \(ss\)-supplement that is a direct summand of \(M\). In this paper, the basic properties and characterizations of \(\oplus_{ss}\)-supplemented modules are provided. In par...
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| Date: | 2024 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2024
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2273 |
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| Journal Title: | Algebra and Discrete Mathematics |
Institution
Algebra and Discrete Mathematics| Summary: | Let \(R\) be a ring and \(M\) be an \(R\)-module. \(M\) is called \(\oplus_{ss}\)-supplemented if every submodule of \(M\) has a \(ss\)-supplement that is a direct summand of \(M\). In this paper, the basic properties and characterizations of \(\oplus_{ss}\)-supplemented modules are provided. In particular, it is shown that \((1)\) if a module \(M\) is \(\oplus_{ss}\)-supplemented, then \(Rad(M)\) is semisimple and \(Soc(M)\unlhd M\); \((2)\) every direct sum of \(ss\)-lifting modules is \(\oplus_{ss}\)-supplemented; \((3)\) a commutative ring \(R\) is an artinian serial ring with semisimple radical if and only if every left \(R\)-module is \(\oplus_{ss}\)-supplemented. |
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