Additive maps of semiprime rings satisfying an Engel condition

Bull. Korean Math. Soc. Published online January 7, 2021

Tsiu-Kwen Lee, Yu Li, and Gaohua Tang
National Taiwan University, Taiwan, Southwest University, Beibu Gulf University, China

Abstract : Let R be a semiprime ring with maximal right ring of quotients Qmr(R), and let n_1, n_2,...,n_k be k fixed positive integers.
Suppose that R is (n_1+n_2+⋯+n_k)!-torsion free, and that f:ρ→Qmr(R) is an additive map, where ρ is a nonzero right ideal of R. It is proved that if [[…[f(x),x^{n_1}],…], x^{n_k}]=0
for all x∈ρ, then [f(x),x]=0 for all x∈ρ.
This gives the result of Beidar et al. \cite{beidar1997} for semiprime rings. Moreover, it is also proved that if R is p-torsion, where p is a prime integer with p=∑ki=1ni, and if f:R→Qmr(R) is an additive map satisfying
[[…[f(x),x^{n_1}],…], x^{n_k}]=0
for all x∈R, then [f(x),x]=0 for all x∈R.

Keywords : Semiprime ring, prime ring, extended centroid, Engel condition, functional identity