A generalized ring homomorphisms equation (Q848878)

Let \(X\) and \(Y\) be two unitary rings. In addition, let \(Y\) be commutative with no zero divisors and \(a, b, c, d, k \in Y\). The authors prove that if the set \(S\) of all solutions \(f: X \to Y\), with \(f(0) = 0\), of the functional equation \[ af(x y) + bf(x) f(y) + cf(x + y) + df(x) + kf(y) = 0 \] is nonempty, then \(k = d = -c\). The main result is a description of the set \(S\) in all possible cases of constants \(a\), \(b\), and \(c\).