Almost f-maps and almost f-rings

Abstract

In the beginning of the eighties, Buskes and Holland proved that any archimedean almost f-ring is commutative. One decade later, Steinberg showed that if G and H are l with H archimedean, then any positive bilinear map G × G $${{\begin{array}{ll} B \\ \rightarrow \end{array}}}$$ H such that $${x \wedge y = 0}$$ implies B(x, y) = 0 is symmetric. At first sight, the Steinberg Theorem might seem to be a considerable generalization of the Buskes and Holland result. It turns out, surprisingly enough, that these two results are equivalent. The main purpose of this paper is to establish this equivalence. A second objective is to apply the aforementioned Steinberg Theorem to prove that if R is an f-ring with a unit element e and S is an archimedean f-ring, then an l-homomorphism R $${{{\begin{array}{ll} h \\\rightarrow \end{array}}}}$$ S is a ring homomorphism if and only if h(e) is idempotent in S. This extends a well-known result by Huijsmans and de Pagter, who obtained the same conclusion for semiprime f-algebras.