# Automorphisms with annihilator condition in prime rings

Keywords:
automorphism, prime ring, generalized polynomial identity

### Abstract

Let*R*be a prime ring,

*I*a nonzero ideal of

*R*, and

*a*∈

*R*. Suppose that

*σ*is a nontrivial automorphism of

*R*such that

*a*{(

*σ*(

*x*∘

*y*))

^{n}− (

*x*∘

*y*)

^{m}} = 0 or

*a*{(

*σ*([

*x*,

*y*]))

^{n}− ([

*x*,

*y*])

^{m}} = 0 for all

*x*,

*y*∈

*I*, where

*n*and

*m*are fixed positive integers. We prove that if char(

*R*) >

*n*+ 1 or char(

*R*) = 0, then either

*a*= 0 or

*R*is commutative.

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