Commutators and products of Lie ideals of prime rings
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912222946525184 |
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| author | Lee, Tsiu-Kwen Lin, Jheng-Huei |
| author_facet | Lee, Tsiu-Kwen Lin, Jheng-Huei |
| contents | Motivated by some recent results on Lie ideals, it is proved that if $L$ is a Lie ideal of a simple ring $R$ with center $Z(R)$, then $L\subseteq Z(R)$, $L=Z(R)a+Z(R)$ for some noncentral $a\in L$, or $[R, R]\subseteq L$, which gives a generalization of a classical theorem due to Herstein. We also study commutators and products of noncentral Lie ideals of prime rings. Precisely, let $R$ be a prime ring with extended centroid $C$. We completely characterize Lie ideals $L$ and elements $a$ of $R$ such that $L+aL$ contains a nonzero ideal of $R$. Given noncentral Lie ideals $K, L$ of $R$, it is proved that $[K, L]=0$ if and only if $KC=LC=Ca+C$ for any noncentral element $a\in L$. As a consequence, we characterize noncentral Lie ideals $K_1,\ldots,K_m$ with $m\geq 2$ such that $K_1K_2\cdots K_m$ contains a nonzero ideal of $R$. Finally, we characterize noncentral Lie ideals $K_j$'s and $L_k$'s satisfying $\big[K_1K_2\cdots K_m, L_1L_2\cdots L_n\big]=0$ from the viewpoint of centralizers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_00444 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Commutators and products of Lie ideals of prime rings Lee, Tsiu-Kwen Lin, Jheng-Huei Rings and Algebras 16N60 (Primary), 16W10 (Secondary) Motivated by some recent results on Lie ideals, it is proved that if $L$ is a Lie ideal of a simple ring $R$ with center $Z(R)$, then $L\subseteq Z(R)$, $L=Z(R)a+Z(R)$ for some noncentral $a\in L$, or $[R, R]\subseteq L$, which gives a generalization of a classical theorem due to Herstein. We also study commutators and products of noncentral Lie ideals of prime rings. Precisely, let $R$ be a prime ring with extended centroid $C$. We completely characterize Lie ideals $L$ and elements $a$ of $R$ such that $L+aL$ contains a nonzero ideal of $R$. Given noncentral Lie ideals $K, L$ of $R$, it is proved that $[K, L]=0$ if and only if $KC=LC=Ca+C$ for any noncentral element $a\in L$. As a consequence, we characterize noncentral Lie ideals $K_1,\ldots,K_m$ with $m\geq 2$ such that $K_1K_2\cdots K_m$ contains a nonzero ideal of $R$. Finally, we characterize noncentral Lie ideals $K_j$'s and $L_k$'s satisfying $\big[K_1K_2\cdots K_m, L_1L_2\cdots L_n\big]=0$ from the viewpoint of centralizers. |
| title | Commutators and products of Lie ideals of prime rings |
| topic | Rings and Algebras 16N60 (Primary), 16W10 (Secondary) |
| url | https://arxiv.org/abs/2410.00444 |