Idempotent identities in f-rings

Abstract

Let A be an Archimedean f-ring with identity and assume that A is equipped with another multiplication \(*\) so that A is an f-ring with identity u. Obviously, if \(*\) coincides with the original multiplication of A then u is idempotent in A (i.e., \(u^{2}=u\)). Conrad proved that the converse also holds, meaning that, it suffices to have \(u^{2}=u\) to conclude that \(*\) equals the original multiplication on A. The main purpose of this paper is to extend this result as follows. Let A be a (not necessary unital) Archimedean f-ring and B be an \(\ell \)-subgroup of the underlaying \(\ell \)-group of A. We will prove that if B is an f-ring with identity u, then the equality \(u^{2}=u\) is a necessary and sufficient condition for B to be an f-subring of A. As a key step in the proof of this generalization, we will show that the set of all f-subrings of A with the same identity has a smallest element and a greatest element with respect to the inclusion ordering. Also, we shall apply our main result to obtain a well known characterization of f-ring homomorphisms in terms of idempotent elements.