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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2508.21465 |
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| _version_ | 1866918132584546304 |
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| author | Bovdi, Victor Zabavsky, Bohdan |
| author_facet | Bovdi, Victor Zabavsky, Bohdan |
| contents | We introduce a concept of rings of right (left) almost stable range $1$ and we construct a theory of a canonical diagonal reduction of matrices over such rings. A description of new classes of noncommutative elementary divisor rings is done as well. In particular, for Bézout $D$-domain we introduced the notions of $D$-adequate element and $D$-adequate ring. We proved that every $D$-adequate Bézout domain has almost stable range $1$. For Hermite $D$-ring we proved the necessary and sufficient conditions to be an elementary divisor ring. A ring $R$ is called an $L$-ring if the condition $RaR = R$ for some $a\in R$ implies that $a$ is a unit of $R$. We proved that every $L$-ring of almost stable range $1$ is a ring of right almost stable range $1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_21465 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Rings of the right (left) almost stable range 1 Bovdi, Victor Zabavsky, Bohdan Rings and Algebras We introduce a concept of rings of right (left) almost stable range $1$ and we construct a theory of a canonical diagonal reduction of matrices over such rings. A description of new classes of noncommutative elementary divisor rings is done as well. In particular, for Bézout $D$-domain we introduced the notions of $D$-adequate element and $D$-adequate ring. We proved that every $D$-adequate Bézout domain has almost stable range $1$. For Hermite $D$-ring we proved the necessary and sufficient conditions to be an elementary divisor ring. A ring $R$ is called an $L$-ring if the condition $RaR = R$ for some $a\in R$ implies that $a$ is a unit of $R$. We proved that every $L$-ring of almost stable range $1$ is a ring of right almost stable range $1$. |
| title | Rings of the right (left) almost stable range 1 |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2508.21465 |