Ring homomorphisms equation revisited

Abstract

The functional equation f(x + y) + f(xy) = f{x) + f(y) + f(x)f(y) (*) has been studied by J. Dhombres (Relations de dépendance entre les équations fonctionnelles de Cauchy, Aequationes Math. 35 (1988), 186- 212) for functions / mapping a given ring into another one. In this paper both rings were supposed to have unit elements; additionally the division by 2 had to be performable. Without these assumptions the study of equation (*) becomes considerably more sophisticated (see author’s paper On an equation of ring homomorphisms, Publ. Math. Debrecen 52 (1998), 397-417). At present, we deal with equation (*) assuming that the domain is a unitary ring with no assumptions whatsoever upon the target ring (except for the second part of Theorem 5).