On Strongly \( J^{\#} \)-Clean Rings
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908500311932928 |
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| author | Danchev, Peter Karamali, Gholamreza Hasanzadeh, Omid Esfandiar, Mehrdad |
| author_facet | Danchev, Peter Karamali, Gholamreza Hasanzadeh, Omid Esfandiar, Mehrdad |
| contents | We define and examine the class of {\it strongly \( J^{\#} \)-clean rings} consisting of those rings $R$ such that each element of $R$ is the sum of an idempotent from $R$ and an element from $J^{\#}(R)$ that commute with each other. More exactly, we prove that these rings are simultaneously strongly clean and Dedekind-finite as well as that they factor-ring modulo the Jacobson radical is always Boolean, and also provide some close relations with certain other well-established classes of rings like these of local, semi-local and strongly J-clean rings (as introduced by Chen on 2010) showing the surprising fact that the classes of strongly \( J^{\#} \)-clean and strongly J-clean rings, actually, do coincide. Moreover, a few more extensions of the newly defined class such as group rings and generalized matrix rings are provided too. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_17368 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Strongly \( J^{\#} \)-Clean Rings Danchev, Peter Karamali, Gholamreza Hasanzadeh, Omid Esfandiar, Mehrdad Rings and Algebras Representation Theory 16N40, 16S50, 16U99 We define and examine the class of {\it strongly \( J^{\#} \)-clean rings} consisting of those rings $R$ such that each element of $R$ is the sum of an idempotent from $R$ and an element from $J^{\#}(R)$ that commute with each other. More exactly, we prove that these rings are simultaneously strongly clean and Dedekind-finite as well as that they factor-ring modulo the Jacobson radical is always Boolean, and also provide some close relations with certain other well-established classes of rings like these of local, semi-local and strongly J-clean rings (as introduced by Chen on 2010) showing the surprising fact that the classes of strongly \( J^{\#} \)-clean and strongly J-clean rings, actually, do coincide. Moreover, a few more extensions of the newly defined class such as group rings and generalized matrix rings are provided too. |
| title | On Strongly \( J^{\#} \)-Clean Rings |
| topic | Rings and Algebras Representation Theory 16N40, 16S50, 16U99 |
| url | https://arxiv.org/abs/2508.17368 |