$f:R \to D$ a homomorphism of the additive group of rings , $f(aba)=f(a)f(b)f(a) , f(1_R)=1_D$ , then is $f$ a ring homomorphism?

Question

Let $R$ be a ring with multiplicative identity $1$ and $D$ be an integral domain with multiplicative identity ( i.e. $D$ is a commutative unital ring without zero divisors ) , let $f:R \to D$ be a function satisfying $f(x+y)=f(x)+f(y) , \forall x,y \in R ; f(aba)=f(a)f(b)f(a) , \forall a,b \in R ; f(1_R)=1_D $ ; then is it true that $f$ is a ring homomorphism i.e is it true that $f(ab)=f(a)f(b) \forall a,b \in R$ ? I only know that $f(a^k)=(f(a))^k , \forall a \in R , k \in \mathbb N$ . Please help.

Answer 1

I. N. Herstein, Jordan Homomorphisms, link

N. Jacobson, C. E. Rickart, Jordan Homomorphisms of rings, link