Notes on symmetric skew $n$-derivation in rings
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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.
Abstract
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.
참고문헌 (14)
- M. Ashraf, "On symmetric bi-derivations in rings", Rend. Istit. Mat. Univ. Trieste, 31(1-2), pp.25~36 (1999)
- M. Ashraf, "Traces of permuting n-additive maps and permuting n-derivations of rings", Mediterr. J. Math, 11(2), pp.287~297 (2014)
- H. E. Bell, "Centralizing mappings of semiprime rings", Canad. Math. Bull, 30(1), pp.92~101 (1987)
- M. Bresar, "On generalized biderivations and related maps", J. Algebra, 172(3), pp.764~786 (1995)
- M. Bresar, "Centralizing maps in prime rings with involution", J. Algebra, 161(2), pp.342~357 (1993)
- Ajda Fošner, "Prime and semiprime rings with symmetric skew 3-derivations", Aequationes mathematicae, 87(1-2), pp.191~200 (2014), Springer Nature[DOI]
- 정용수, "On prime and semiprime rings with permuting 3-derivations", 대한수학회보, 44(4), pp.789~794 (2007), 대한수학회
- Gy. Maksa, "A remark on symmetric biadditive functions having nonnegative diagonal-ization", Glas. Mat. Ser. III, 15(35)통권 : 2, pp.279~282 (1980)
- Gy. Maksa, "On the trace of symmetric bi-derivations", C. R. Math. Rep. Acad. Sci. Canada, 9(6), pp.303~307 (1987)
- 박규홍, "ON PRIME AND SEMIPRIME RINGS WITH SYMMETRIC n-DERIVATIONS", 충청수학회지, 22(3), pp.451~458 (2009), 충청수학회
- E. C. Posner, "Derivations in prime rings", Proc. Amer. Math. Soc, 8, pp.1093~1100 (1957)
- R. K. Sharma, "Skew-commuting and commuting mappings in rings with left identity", Results Math, 46(1-2), pp.123~129 (2004)
- J. Vukman, "Symmetric bi-derivations on prime and semi-prime rings", Aequationes Math, 38(2-3), pp.245~254 (1989)
- J. Vukman, "Two results concerning symmetric bi-derivations on prime rings", Aequationes Math, 40(2-3), pp.181~189 (1990)