Axiom 3 modules

Hill, Paul; Megibben, Charles

doi:10.1090/S0002-9947-1986-0833705-6

Axiom $3$ modules
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by Paul Hill and Charles Megibben PDF

Trans. Amer. Math. Soc. 295 (1986), 715-734 Request permission

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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