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- Title
On n-generalized commutators and Lie ideals of rings.
- Authors
Danchev, Peter V.; Lee, Tsiu-Kwen
- Abstract
Let R be an associative ring. Given a positive integer n ≥ 2 , for a 1 , ... , a n ∈ R we define [ a 1 , ... , a n ] n : = a 1 a 2 ⋯ a n − a n a n − 1 ⋯ a 1 , the n -generalized commutator of a 1 , ... , a n . 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 [ x 1 , ... , x r , a , y 1 , ... , y s ] n ∈ A for all x i , y j ∈ 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.
- Subjects
PORTUGAL; COMMUTATION (Electricity); ASSOCIATIVE rings; COMMUTATORS (Operator theory); NONCOMMUTATIVE rings
- Publication
Journal of Algebra & Its Applications, 2022, Vol 21, Issue 11, p1
- ISSN
0219-4988
- Publication type
Article
- DOI
10.1142/S0219498822502218