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Let $R$ be a prime ring (or semiprime ring) with center $Z(R)$, $I$ a nonzero ideal of $R,$ $T$ an automorphism of $R$, $S:R^{n}\rightarrow R$ be a symmetric skew $n$-derivation associated with the automorphism $T$ and $ \Delta $ is the trace of $S.$ In this paper, we shall prove that $ S(x_{1},\ldots ,x_{n})=0$ for all $x_{1},\ldots ,x_{n}\in R$ if any one of the following holds: i) $\Delta (x)=0,$ ii) $[\Delta (x),T(x)]=0$ for all $ x\in I.$ Moreover, we prove that if $[\Delta (x),T(x)]\in Z(R)$ for all $x\in I,$ then $R$ is a commutative ring.
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참고문헌 (14)
참고문헌 신청
M. Ashraf / 1999 / On symmetric bi-derivations in rings / Rend. Istit. Mat. Univ. Trieste 31(1-2):25~36
M. Ashraf / 2014 / Traces of permuting n-additive maps and permuting n-derivations of rings / Mediterr. J. Math 11(2):287~297
H. E. Bell / 1987 / Centralizing mappings of semiprime rings / Canad. Math. Bull 30(1):92~101
M. Bresar / 1995 / On generalized biderivations and related maps / J. Algebra 172(3):764~786
M. Bresar / 1993 / Centralizing maps in prime rings with involution / J. Algebra 161(2):342~357
M. Ashraf / 2014 / Traces of permuting n-additive maps and permuting n-derivations of rings / Mediterr. J. Math 11(2):287~297
H. E. Bell / 1987 / Centralizing mappings of semiprime rings / Canad. Math. Bull 30(1):92~101
M. Bresar / 1995 / On generalized biderivations and related maps / J. Algebra 172(3):764~786
M. Bresar / 1993 / Centralizing maps in prime rings with involution / J. Algebra 161(2):342~357
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