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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Lattice-ordered injective hulls

Author: Stuart A. Steinberg
Journal: Trans. Amer. Math. Soc. 169 (1972), 365-388
MSC: Primary 06A55
MathSciNet review: 0313158
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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.

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Keywords: $ f$-module, lattice-ordered injective hull, relative injectives, torsion-free $ f$-module, Hahn embedding theorem
Article copyright: © Copyright 1972 American Mathematical Society