Antiautomorphisms with quasi-generalized Engel condition

作者：Chacron M^*

来源：Journal of Algebra and Its Applications, 2018, 17(8): 1850145.

DOI：10.1142/S0219498818501451

• Antiautomorphism
• commuting map
• quasi-generalized Engel condition
• involution
• prime ring
• strongly hyperbolic antiautomorphism

Let R be a ring with 1. Given elements a, b of R and the integer n >= 1 define [a, b](0) := a, [a, b](1) := [a, b](= ab - ba) and [a, b](n) := [[a, b](n-1), b]. We say that a given antiautomorphism mu of R is commuting if [mu(x), x(m(x))] = 0, all x is an element of R. More generally, assume that mu satisfies the condition [mu(x), x(m(x))](n(x)) = 0 where m(x), n(x) are corresponding positive integers depending on x, and x ranges over R. To what extent can one say that mu is commuting? In this paper, we answer the question in the affirmative if R is a prime ring containing some idempotent element e = e(2) not equal 0, 1. In the diametrically opposed case in which R is a division ring the answer is again yes provided R is algebraic over its center and mu is of finite order. These two major complementary results will be put to work to provide an answer to the general question.

• 出版日期2018-8