On Lie isomorphisms of rings
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912247659364352 |
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| author | Bezushchak, Oksana Kashuba, Iryna Zelmanov, Efim |
| author_facet | Bezushchak, Oksana Kashuba, Iryna Zelmanov, Efim |
| contents | An associative ring $A$ gives rise to the Lie ring $A^{(-)}=(A,[a,b ]=ab-ba)$. The subject of isomorphisms of Lie rings $A^{(-)}$ and $[A,A]$ has attracted considerable attention in the literature. We prove that if the identity element of $A$ decomposes into a sum of at least three full orthogonal idempotents, then any isomorphism from the Lie ring $[A,A]$ to the Lie ring $[B,B]$ is standard.
For non-unital rings, the description is more intricate. Under a certain assumption on idempotents, we extend a Lie isomorphism from $[A,A]$ to $[B,B]$ to a homomorphism of associative rings $\widehat{A\oplus A^{op}}\to B,$ where $A^{op}=(A,a\cdot b= b\cdot a),$ and $\widehat{A\oplus A^{op}}\to A\oplus A^{op}$ is the universal annihilator extension of the ring $A\oplus A^{op}.$
The results obtained are then applied to the description of automorphisms and derivations of Lie algebras of infinite matrices. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_18903 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Lie isomorphisms of rings Bezushchak, Oksana Kashuba, Iryna Zelmanov, Efim Rings and Algebras Primary 15B30, 16W10, 17B60, Secondary 16W20, 17B40 An associative ring $A$ gives rise to the Lie ring $A^{(-)}=(A,[a,b ]=ab-ba)$. The subject of isomorphisms of Lie rings $A^{(-)}$ and $[A,A]$ has attracted considerable attention in the literature. We prove that if the identity element of $A$ decomposes into a sum of at least three full orthogonal idempotents, then any isomorphism from the Lie ring $[A,A]$ to the Lie ring $[B,B]$ is standard. For non-unital rings, the description is more intricate. Under a certain assumption on idempotents, we extend a Lie isomorphism from $[A,A]$ to $[B,B]$ to a homomorphism of associative rings $\widehat{A\oplus A^{op}}\to B,$ where $A^{op}=(A,a\cdot b= b\cdot a),$ and $\widehat{A\oplus A^{op}}\to A\oplus A^{op}$ is the universal annihilator extension of the ring $A\oplus A^{op}.$ The results obtained are then applied to the description of automorphisms and derivations of Lie algebras of infinite matrices. |
| title | On Lie isomorphisms of rings |
| topic | Rings and Algebras Primary 15B30, 16W10, 17B60, Secondary 16W20, 17B40 |
| url | https://arxiv.org/abs/2502.18903 |