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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Lattice-ordered injective hulls
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by Stuart A. Steinberg PDF
Trans. Amer. Math. Soc. 169 (1972), 365-388 Request permission

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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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 169 (1972), 365-388
  • MSC: Primary 06A55
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0313158-7
  • MathSciNet review: 0313158