Lie ideals and derivations of exceptional prime rings
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866913972508164096 |
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| author | Lee, Tsiu-Kwen |
| author_facet | Lee, Tsiu-Kwen |
| contents | A prime ring $R$ with extended centroid $C$ is said to be exceptional if both $\text{\rm char}\,R=2$ and $\dim_CRC=4$. Herstein characterized additive subgroups $A$ of a nonexceptional simple ring $R$ satisfying $\big[A, [R, R]\big]\subseteq A$. In 1972 Lanski and Montgomery extended Herstein's theorem to nonexceptional prime rings. In the paper we first extend Herstein's theorem to arbitrary simple rings. For the prime case, let $R$ be an exceptional prime ring with center $Z(R)$. It is proved that if $A$ is a noncentral additive subgroup of $R$ satisfying $\big[A, L\big]\subseteq A$ for some nonabelian Lie ideal $L$ of $R$, then $βZ(R)\subseteq A$ for some nonzero $β\in Z(R)$, and either $AC=Ca+C$ for some $a\in A\setminus Z(R)$ with $a^2\in Z(R)$ or $[RC, RC]\subseteq AC$. Secondly, we study certain generalized linear identities satisfied by Lie ideals and then completely characterize derivations $δ, d$ of $R$ satisfying $δd(L)\subseteq Z(R)$ for $L$ a Lie ideal of $R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_01544 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Lie ideals and derivations of exceptional prime rings Lee, Tsiu-Kwen Rings and Algebras 16N60 A prime ring $R$ with extended centroid $C$ is said to be exceptional if both $\text{\rm char}\,R=2$ and $\dim_CRC=4$. Herstein characterized additive subgroups $A$ of a nonexceptional simple ring $R$ satisfying $\big[A, [R, R]\big]\subseteq A$. In 1972 Lanski and Montgomery extended Herstein's theorem to nonexceptional prime rings. In the paper we first extend Herstein's theorem to arbitrary simple rings. For the prime case, let $R$ be an exceptional prime ring with center $Z(R)$. It is proved that if $A$ is a noncentral additive subgroup of $R$ satisfying $\big[A, L\big]\subseteq A$ for some nonabelian Lie ideal $L$ of $R$, then $βZ(R)\subseteq A$ for some nonzero $β\in Z(R)$, and either $AC=Ca+C$ for some $a\in A\setminus Z(R)$ with $a^2\in Z(R)$ or $[RC, RC]\subseteq AC$. Secondly, we study certain generalized linear identities satisfied by Lie ideals and then completely characterize derivations $δ, d$ of $R$ satisfying $δd(L)\subseteq Z(R)$ for $L$ a Lie ideal of $R$. |
| title | Lie ideals and derivations of exceptional prime rings |
| topic | Rings and Algebras 16N60 |
| url | https://arxiv.org/abs/2508.01544 |