Symmetry of Syzygies of a System of Functional Equations Defining a Ring Homomorphism

Abstract

I deal with an alienation problem for the system of two fundamental Cauchy functional equations with an unknown function f mapping a ring X into an integral domain Y and preserving binary operations of addition and multiplication, respectively. The resulting syzygies obtained by adding (resp. multiplying) these two equations side by side are discussed. The first of these two syzygies was first examined by Jean Dhombres in 1988 who proved that under some additional conditions concering the domain and range rings it forces f to be a ring homomorphism (alienation phenomenon). The novelty of the present paper is to look for sufficient conditions upon f solving the other syzygy to be alien.