Lattice-ordered injective hulls

Abstract:

It is well known that the injective hull of a lattice-ordered group ($l$-group) $M$ can be given a lattice order in a unique way so that it becomes an $l$-group extension of $M$. This is not the case for an arbitrary $f$-module over a partially ordered ring (po-ring). The fact that it is the case for any $l$-group is used extensively to get deep theorems in the theory of $l$-groups. For instance, it is used in the proof of the Hahn-embedding theorem and in the characterization of ${\aleph _a}$-injective $l$-groups. In this paper we give a necessary and sufficient condition on the injective hull of a torsion-free $f$-module $M$ (over a directed essentially positive po-ring) for it to be made into an $f$-module extension of $M$ (in a unique way). An $f$-module is called an $i - f$-module if its injective hull can be made into an $f$-module extension. The class of torsion-free $i - f$-modules is closed under the formation of products, sums, and Hahn products of strict $f$-modules. Also, an $l$-submodule and a torsion-free homomorphic image of a torsion-free $i - f$-module are $i - f$-modules. Let $R$ be an $f$-ring with zero right singular ideal whose Boolean algebra of polars is atomic. We show that $R$ is a $qf$-ring (i.e., ${R_R}$ is an $i - f$-module) if and only if each torsion-free $R - f$-module is an $i - f$-module. There are no injectives in the category of torsion-free $R - f$-modules, but there are ${\aleph _a}$-injectives. These may be characterized as the $f$-modules that are injective $R$-modules and ${\aleph _a}$-injective $l$-groups. In addition, each torsion-free $f$-module over $R$ can be embedded in a Hahn product of $l$-simple $Q(R) - f$-modules. We note, too, that a totally ordered domain has an $i - f$-module if and only if it is a right Ore domain.