On generalized left derivations in rings and Banach algebras. (Q543288)

The author considers consequences of the existence of certain special maps of semiprime rings. Let \(R\) be a 2-torsion free semiprime ring with additive \(G,d\colon R\to R\) satisfying \(d(x^2)=2xd(x)\) and \(G(x^2)=xG(x)+xd(x)\) for all \(x\in R\). The first theorems of the paper show that \(d\) is a derivation of \(R\) with \(d(R)\) central, and then that for all \(x,y\in R\), \(G(xy)=xG(y)+yG(x)\). When \(R\) is a noncommutative prime ring with \(\text{char\,}R\neq 2\) and Martindale quotient ring \(Q\), and when \(G\) acts as a homomorphism (anti-homomorphism) on a nonzero \(I\) of \(R\), then there is \(q\in Q\) so that \(G(r)=rq\) for all \(r\in R\). The next theorem assumes that \(R\) is any noncommutative prime ring and that \(H,F\colon R\times R\to R\) are biadditive and satisfy \(F(xy,z)=xF(y,z)+yF(x,z)\) and \(H(xy,z)=xH(y,z)+yF(x,z)\). The conclusion is that \(F(R\times R)=0\). The final theorem in the paper assumes that \(R\) is a semisimple Banach algebra with \(G\) and \(d\) as above and shows that \(G\) must be continuous.