Additivity and exponentiality are alien to each other

Let (S, +) be a (semi)group and let (R,+, .) be an integral domain. We study the solutions of a Pexider type functional equation f(x + y) + g(x + y) = f(x) + f(y) + g(x) g(y) for functions f and g mapping S into R. Our chief concern is to examine whether or not this functional equation is equivalent to the system of two Cauchy equations {f(x + y) = f(x) + f(y) g(x + y) = g(x) g(y) for every x, y is an element of S.