Geodesic
Derivations with Engel conditions in prime and semiprime rings
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1135-1140
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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.
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.
DOI : 10.1007/s10587-011-0053-7
Classification : 16N60, 16R50, 16U70, 16U80, 16W25
Keywords: prime and semiprime rings; ideal; derivation; GPIs 
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Huang, Shuliang. Derivations with Engel conditions in prime and semiprime rings. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1135-1140. doi: 10.1007/s10587-011-0053-7

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