Let $R$ be a prime ring with center $Z(R)$, and $d$ a derivation of $R$. Suppose that $(d[x, y]_k)^n-m[x, y]_kin Z(R)$ for all $x, y in R$, where $meq n, k geq 1$ are fixed integers. Then $d=0$ or $R$ satisfies $s_4$, the standard identity in four variables. In the case $(d[x, y]_k)^n-m[x, y]_k=0$ for all $x, y in R$, then $d=0$ or $R$ is commutative.
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机译:假设$ R $是中心为ZZR的素环,而$ d $是R的导数。假设$(d [x,y] _k)^ nm [x,y] _k in Z(R)$对于所有$ x,y in R $,其中$ m neq n,k geq 1 $是固定的整数。然后$ d = 0 $或$ R $满足$ s_4 $,这是四个变量中的标准身份。在$(d [x,y] _k)^ n-m [x,y] _k = 0 $的情况下,R $中的所有$ x,y n,则$ d = 0 $或$ R $是可交换的。
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