Additivity of multiplicative (generalized) maps over rings

Abstract: In this paper, we prove that a bijective map $\varphi$ over a ring $R$ with a non-trivial idempotent satisfying $\varphi(ab)=\varphi(a)\varphi(b),$ for all $a,b\in R$, is additive. Also, we prove that a map $D$ on $R$ satisfying $D(ab)=D(a)b+\varphi(a) D(b),$ for all $a,b\in R$ where $\varphi$ is the map just mentioned above, is additive. Moreover, we establish that if a map $g$ over $R$ satisfies $g(ab)=g(a)b+\varphi(a)D(b),$ for all $a,b\in R$ and above mentioned maps $\varphi$ and $D$, then $g$ is additive.