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Transactions of the American Mathematical Society

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



Axiom $3$ modules

Authors: Paul Hill and Charles Megibben
Journal: Trans. Amer. Math. Soc. 295 (1986), 715-734
MSC: Primary 20K21; Secondary 20K10
MathSciNet review: 833705
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Abstract: By introducing the concept of a knice submodule, a refinement of the notion of nice subgroup, we are able to formulate a version of the third axiom of countability appropriate to the study of $p$-local mixed groups in the spirit of the well-known characterization of totally projective $p$-groups. Our Axiom $3$ modules, in fact, form a class of ${{\mathbf {Z}}_{\mathbf {p}}}$-modules, encompassing the totally projectives in the torsion case, for which we prove a uniqueness theorem and establish closure under direct summands. Indeed Axiom $3$ modules turn out to be precisely the previously classified Warfield modules. But with the added power of the third axiom of countability characterization, we derive numerous new results, including the resolution of a long-standing problem of Warfield and theorems in the vein of familiar criteria due to Kulikov and Pontryagin.

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Keywords: Third axiom of countability, knice submodules, Ulm-Kaplansky invariants, Warfield modules, decomposition basis, valuated coproduct, simply presented groups, primitive element
Article copyright: © Copyright 1986 American Mathematical Society