Article
MSC:
16N60,
16R50,
16U70,
16U80,
16W25 | MR 2886261 | Zbl 1240.16048 | DOI: 10.1007/s10587-011-0053-7
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Keywords:
prime and semiprime rings; ideal; derivation; GPIs
prime and semiprime rings; ideal; derivation; GPIs
Summary:
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 $\mathop {\rm Char}R\neq 2$ 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.
References:
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