Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1135-1140, 2011

Derivations with Engel conditions in prime and semiprime rings

Shuliang Huang

Shuliang Huang, Department of Mathematics, Chuzhou University, Chuzhou 239012, P. R. China, e-mail: shulianghuang@sina.com

Abstract: Let $R$ be a prime ring, $I$ a nonzero ideal of $R$, $d$ a derivation of $R$ and $m, n$ fixed positive integers. (i) If $(d[x,y])^m=[x,y]_n$ for all $x,y\in I$, then $R$ is commutative. (ii) If $\mathop CharR\neq2$ and $[d(x),d(y)]_m=[x,y]^n$ for all $x,y\in I$, then $R$ is commutative. Moreover, we also examine the case when $R$ is a semiprime ring.

Keywords: prime and semiprime rings, ideal, derivation, GPIs

Classification (MSC 2010): 16N60, 16U80, 16W25

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