The properties of Jordan derivations of semiprime rings and Banach algebras, I
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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 $A.$
Abstract
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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