https://scholars.lib.ntu.edu.tw/handle/123456789/606439
標題: | On n -generalized commutators and Lie ideals of rings | 作者: | Danchev P.V Lee T.-K. TSIU-KWEN LEE |
關鍵字: | GPI;Idempotent;N -generalized commutator (Lie ideal);Noncommutative polynomial;PI;Prime ring;Regular ring | 公開日期: | 2021 | 來源出版物: | Journal of Algebra and its Applications | 摘要: | Let R be an associative ring. Given a positive integer n ? 2, for a1,...,an R we define [a1,...,an]n:= a1a2?an - anan-1?a1, the n-generalized commutator of a1,...,an. By an n-generalized Lie ideal of R (at the (r + 1)th position with r ? 0) we mean an additive subgroup A of R satisfying [x1,...,xr,a,y1,...,ys]n A for all xi,yj R and all a A, where r + s = n - 1. In the paper, we study n-generalized commutators of rings and prove that if R is a noncommutative prime ring and n ? 3, then every nonzero n-generalized Lie ideal of R contains a nonzero ideal. Therefore, if R is a noncommutative simple ring, then R = [R,...,R]n. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137-139]. Some generalizations and related questions on n-generalized commutators and their relationship with noncommutative polynomials are also discussed. ? 2022 World Scientific Publishing Company. |
URI: | https://www.scopus.com/inward/record.uri?eid=2-s2.0-85113249239&doi=10.1142%2fS0219498822502218&partnerID=40&md5=789d16df9a5ae3222b33791efe9f9692 https://scholars.lib.ntu.edu.tw/handle/123456789/606439 |
ISSN: | 02194988 | DOI: | 10.1142/S0219498822502218 |
顯示於: | 數學系 |
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