A note on additive commutator groups in certain algebras
Fuente:
arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| author | Ha, Nguyen Thi Thai Son, Tran Nam Vinh, Pham Duy |
| author_facet | Ha, Nguyen Thi Thai Son, Tran Nam Vinh, Pham Duy |
| contents | We study whether a unital associative algebra $ A $ over a field admits a decomposition of the form $A = Z(A) + [A,A]$ where $ Z(A) $ is the center of $ A $ and $ [A,A] $ denotes the additive subgroup of $A$ generated by all additive commutators of $A$. Among our main considerations are the cases in which $A$ is the matrix ring over a division ring, a generalized quaternion algebra, or a semisimple finite-dimensional algebra. We also discuss some applications that do not necessarily require the decomposition, such as the case where $ A $ is the twisted group algebra of a locally finite group over a field of characteristic zero: if all additive commutators of $A$ are central, then $ A $ must be commutative. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_13303 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A note on additive commutator groups in certain algebras Ha, Nguyen Thi Thai Son, Tran Nam Vinh, Pham Duy Rings and Algebras We study whether a unital associative algebra $ A $ over a field admits a decomposition of the form $A = Z(A) + [A,A]$ where $ Z(A) $ is the center of $ A $ and $ [A,A] $ denotes the additive subgroup of $A$ generated by all additive commutators of $A$. Among our main considerations are the cases in which $A$ is the matrix ring over a division ring, a generalized quaternion algebra, or a semisimple finite-dimensional algebra. We also discuss some applications that do not necessarily require the decomposition, such as the case where $ A $ is the twisted group algebra of a locally finite group over a field of characteristic zero: if all additive commutators of $A$ are central, then $ A $ must be commutative. |
| title | A note on additive commutator groups in certain algebras |
| topic | Rings and Algebras |
| url | https://arxiv.org/abs/2505.13303 |