A GEOMETRIC CHARACTERIZATION OF RING HOMOMORPHISMS ON f-RINGS

Abstract

Let A be an f-ring with identity u and B be an archimedean f-ring. For every idempotent element w in B, let [Formula: see text] denote the set of all positive group homomorphisms ℌ : A → B with ℌ(u) = w. We prove that [Formula: see text] is a ring homomorphism if and only if ℌ is an extreme point of [Formula: see text]. As a consequence, we derive a characterization of ring homomorphisms in [Formula: see text] in terms of a Gelfand-type transform. Moreover, we show that ring homomorphisms in [Formula: see text] are, up to multiplicative constants, all the basic elements of the ℓ-group of all bounded group homomorphisms from A into ℝ.