Unit Uniquely Clean Rings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916960416038912 |
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| author | Doostalizadeh, Mina Moussavi, Ahmad Danchev, Peter |
| author_facet | Doostalizadeh, Mina Moussavi, Ahmad Danchev, Peter |
| contents | We define the class of {\it unit uniquely clean} rings ({\it UnitUC} for short), that is a common generalization of uniquely clean rings and strongly nil clean rings. Abelian {\it UnitUC} rings are uniquely clean and {\it UnitUC} rings with nil Jacobson radical are strongly nil clean. These rings also generalize the UUC and CUC rings, defined by Calugareanu-Zhou in Mediterranean J. Math. (2023), which are rings whose clean elements are uniquely clean. These rings are also represent a natural generalization of the Boolian rings in that a ring is {\it UnitUC} if, and only if, it is exchange and Boolean modulo the Jacobson radical. The behavior of {\it UnitUC} rings under group ring and matrix ring extensions is investigated. Several examples are provided to explain and delimit the results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_17573 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unit Uniquely Clean Rings Doostalizadeh, Mina Moussavi, Ahmad Danchev, Peter Rings and Algebras Representation Theory 16S34, 16U60, 20C07 We define the class of {\it unit uniquely clean} rings ({\it UnitUC} for short), that is a common generalization of uniquely clean rings and strongly nil clean rings. Abelian {\it UnitUC} rings are uniquely clean and {\it UnitUC} rings with nil Jacobson radical are strongly nil clean. These rings also generalize the UUC and CUC rings, defined by Calugareanu-Zhou in Mediterranean J. Math. (2023), which are rings whose clean elements are uniquely clean. These rings are also represent a natural generalization of the Boolian rings in that a ring is {\it UnitUC} if, and only if, it is exchange and Boolean modulo the Jacobson radical. The behavior of {\it UnitUC} rings under group ring and matrix ring extensions is investigated. Several examples are provided to explain and delimit the results. |
| title | Unit Uniquely Clean Rings |
| topic | Rings and Algebras Representation Theory 16S34, 16U60, 20C07 |
| url | https://arxiv.org/abs/2509.17573 |