Almost f-maps and almost f-rings

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.

Authors and Affiliations

1. Département du Cycle Arégatif, Insitut Préparatoire aux Études Scientifiques et Techniques, Université de Carthage, B.P. 51 - 2070, La Marsa, Tunisia

   Mohamed Amine Ben Amor

2. Département de Mathématiques, Faculté des Sciences Mathématiques, Physiques et Naturelles de Tunis, Université Tunis-El Manar, 2092, Tunis, Tunisia

   Karim Boulabiar

Corresponding author

Correspondence to Mohamed Amine Ben Amor.

Presented by W. McGovern.

The authors acknowledge support from Research Laboratory LATAO Grant LR11ES12.

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