Systems of Inequalities Characterizing Ring Homomorphisms

Abstract

Assume that T:P→R and U:P→R are arbitrary mappings between two partially ordered rings P and R. We study a few systems of functional inequalities which characterize ring homomorphisms. For example, we prove that if T and U satisfy T(f+g)≥T(f)+T(g), U(f·g)≥U(f)·U(g), for all f,g∈P and T≥U, then U=T and this mapping is a ring homomorphism. Moreover, we find two other systems for which we obtain analogous assertions.