example of a map in a prime ring

Question

Let $R$ be a prime ring, and $f: R\mapsto R~$ an additive map such that $$f(xy)=f(x)f(y)+f(x)y+xf(y)\quad \forall~x,y\in R $$ It is clear that if we take any endomorphism $g$ of R then $g-id_R$ verify the above formula.
My question is : can we provide another example ?

Answer 1

Hint: To show that given any such map $f$, it "comes from" such a ring endomorphism $g$, why not set $g := id_R + f$ and try to show that $g$ a ring endomorphism of $R$?