Strongly clean ring elements that are one-sided inverses
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913998238121984 |
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| author | Bergman, George M. |
| author_facet | Bergman, George M. |
| contents | A longstanding open question is whether every strongly clean ring (ring in which every element is strongly clean, i.e., is the sum of an idempotent and a unit which commute with each other) is Dedekind-finite (has the property that every element with a one-sided inverse is invertible). We give an example of a ring with two strongly clean elements that are one-sided, but not two-sided, inverses of one another, suggesting that the answer to that question may be negative.
We then discuss possible ways of strengthening this result to give a full negative answer. We end with some brief observations on related topics, in particular, uniquely strongly clean rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_14396 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strongly clean ring elements that are one-sided inverses Bergman, George M. Rings and Algebras 16U40, 16U60 (Primary) 16S50, 16U90. (Secondary) A longstanding open question is whether every strongly clean ring (ring in which every element is strongly clean, i.e., is the sum of an idempotent and a unit which commute with each other) is Dedekind-finite (has the property that every element with a one-sided inverse is invertible). We give an example of a ring with two strongly clean elements that are one-sided, but not two-sided, inverses of one another, suggesting that the answer to that question may be negative. We then discuss possible ways of strengthening this result to give a full negative answer. We end with some brief observations on related topics, in particular, uniquely strongly clean rings. |
| title | Strongly clean ring elements that are one-sided inverses |
| topic | Rings and Algebras 16U40, 16U60 (Primary) 16S50, 16U90. (Secondary) |
| url | https://arxiv.org/abs/2508.14396 |