Prime rings having nontrivial centralizers of (skew) traces of Lie ideals
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915316366311424 |
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| author | Lee, Tsiu-Kwen Lin, Jheng-Huei |
| author_facet | Lee, Tsiu-Kwen Lin, Jheng-Huei |
| contents | Let $R$ be a prime ring with center $Z(R)$ and with involution $*$. Given an additive subgroup $A$ of $R$, let $T(A):=\{x+x^*\mid x\in A\}$ and $K_0(A):=\{x-x^*\mid x\in A\}$. Let $L$ be a non-abelian Lie ideal of $R$. It is proved that if $d$ is a nonzero derivation of $R$ satisfying $d(T(L))=0$ (resp. $d(K_0(L))=0$), then $T(R)^2\subseteq Z(R)$ (resp. $K_0(R)^2\subseteq Z(R)$). These results are applied to the study of $d(T(M))=0$ and $d(K_0(M))=0$ for noncentral $*$-subrings $M$ of a division ring $R$ such that $M$ is invariant under all inner automorphisms of $R$, and for noncentral additive subgroups $M$ of a prime ring $R$ containing a nontrivial idempotent such that $M$ is invariant under all special inner automorphisms of $R$. The obtained theorems also generalize some recent results on simple artinian rings with involution due to M. Chacron. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_05588 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Prime rings having nontrivial centralizers of (skew) traces of Lie ideals Lee, Tsiu-Kwen Lin, Jheng-Huei Rings and Algebras 16N60 (Primary) 16W10, 16K40 (Secondary) Let $R$ be a prime ring with center $Z(R)$ and with involution $*$. Given an additive subgroup $A$ of $R$, let $T(A):=\{x+x^*\mid x\in A\}$ and $K_0(A):=\{x-x^*\mid x\in A\}$. Let $L$ be a non-abelian Lie ideal of $R$. It is proved that if $d$ is a nonzero derivation of $R$ satisfying $d(T(L))=0$ (resp. $d(K_0(L))=0$), then $T(R)^2\subseteq Z(R)$ (resp. $K_0(R)^2\subseteq Z(R)$). These results are applied to the study of $d(T(M))=0$ and $d(K_0(M))=0$ for noncentral $*$-subrings $M$ of a division ring $R$ such that $M$ is invariant under all inner automorphisms of $R$, and for noncentral additive subgroups $M$ of a prime ring $R$ containing a nontrivial idempotent such that $M$ is invariant under all special inner automorphisms of $R$. The obtained theorems also generalize some recent results on simple artinian rings with involution due to M. Chacron. |
| title | Prime rings having nontrivial centralizers of (skew) traces of Lie ideals |
| topic | Rings and Algebras 16N60 (Primary) 16W10, 16K40 (Secondary) |
| url | https://arxiv.org/abs/2412.05588 |