The $X$-semiprimeness of Rings
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911832191533056 |
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| author | Călugăreanu, Grigore Lee, Tsiu-Kwen Matczuk, Jerzy |
| author_facet | Călugăreanu, Grigore Lee, Tsiu-Kwen Matczuk, Jerzy |
| contents | For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and unit-semiprime rings. Secondly, given a Lie ideal $L$ of a ring $R$, we offer a criterion for $R$ to be $L$-semiprime. For a prime ring $R$, we characterizes Lie ideals $L$ of $R$ such that $R$ is $L$-semiprime. Moreover, $X$-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent), semiprime rings, regular rings, and subdirect products are studied. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_19374 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $X$-semiprimeness of Rings Călugăreanu, Grigore Lee, Tsiu-Kwen Matczuk, Jerzy Rings and Algebras 16N60, 16U10, 16U40, 16S50 For a nonempty subset $X$ of a ring $R$, the ring $R$ is called $X$-semiprime if, given $a\in R$, $aXa=0$ implies $a=0$. This provides a proper class of semiprime rings. First, we clarify the relationship between idempotent semiprime and unit-semiprime rings. Secondly, given a Lie ideal $L$ of a ring $R$, we offer a criterion for $R$ to be $L$-semiprime. For a prime ring $R$, we characterizes Lie ideals $L$ of $R$ such that $R$ is $L$-semiprime. Moreover, $X$-semiprimeness of matrix rings, prime rings (with a nontrivial idempotent), semiprime rings, regular rings, and subdirect products are studied. |
| title | The $X$-semiprimeness of Rings |
| topic | Rings and Algebras 16N60, 16U10, 16U40, 16S50 |
| url | https://arxiv.org/abs/2402.19374 |