Additivity of multiplicative (generalized) maps over rings

In this paper, we mainly prove some results on the additivity of maps over rings under certain conditions. First, we discuss a special case of MARTINDALE III's theorem of \cite{1969M} as 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. Then 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 mentioned above, is additive. Finally, 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 the maps $\varphi$ and $D$ are mentioned above, then $g$ is additive.