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Let $R$ be a $5!$-torsion free semiprime ring, and let $D:R\to R$ be a Jordan derivation on a semiprime ring $R.$ Then $[D(x),x]D(x)^2=0$ if and only if $D(x)^2[D(x),x]=0$ for every $x\in R.$ In particular, let $A$ be a Banach algebra with $\mbox{rad}(A)$ and if $D$ is a continuous linear Jordan derivation on $A,$ then we show that $[D(x),x]D(x)^2\in \mbox{rad}(A)$ if and only if $D(x)^2[D(x),x]\in \mbox{rad}(A)$ for all $x\in A$ where $\mbox{rad}(A)$ is the Jacobson radical of $A.$
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참고문헌 (17)
참고문헌 신청
F. F. Bonsall / 1973 / Complete Normed Algebras
M. Bresar / 1994 / Derivations of noncommutative Banach algebras II / Arch. Math. 63(1):56~59
M. Bresar / 1988 / Jordan derivations on semiprime rings / Proc. Amer. Math. Soc. 104(4):1003~1006
L. O. Chung / 1981 / Semiprime rings with nilpotent derivatives / Canad. Math. Bull. 24(4):415~421
B. E. Johnson / 1968 / Continuity of derivations and a problem of Kaplansky / Amer. J. Math. 90:1067~1073
M. Bresar / 1994 / Derivations of noncommutative Banach algebras II / Arch. Math. 63(1):56~59
M. Bresar / 1988 / Jordan derivations on semiprime rings / Proc. Amer. Math. Soc. 104(4):1003~1006
L. O. Chung / 1981 / Semiprime rings with nilpotent derivatives / Canad. Math. Bull. 24(4):415~421
B. E. Johnson / 1968 / Continuity of derivations and a problem of Kaplansky / Amer. J. Math. 90:1067~1073
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