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

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

DOI: https://doi.org/10.1090/S0002-9947-1986-0833705-6

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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.

References

László Fuchs, Infinite abelian groups. Vol. II, Pure and Applied Mathematics. Vol. 36-II, Academic Press, New York-London, 1973. MR 0349869
Phillip A. Griffith, Infinite abelian group theory, University of Chicago Press, Chicago, Ill.-London, 1970. MR 0289638

On the classification of abelian groups

On the freeness of abelian groups, a generalization of Pontryagin’s theorem

73

Paul Hill, The third axiom of countability for abelian groups, Proc. Amer. Math. Soc. 82 (1981), no. 3, 347–350. MR 612716, DOI 10.1090/S0002-9939-1981-0612716-0
Paul Hill, Criteria for total projectivity, Canadian J. Math. 33 (1981), no. 4, 817–825. MR 634140, DOI 10.4153/CJM-1981-063-x
Paul Hill, Isotype subgroups of totally projective groups, Abelian group theory (Oberwolfach, 1981) Lecture Notes in Math., vol. 874, Springer, Berlin-New York, 1981, pp. 305–321. MR 645937
Roger Hunter, Fred Richman, and Elbert Walker, Warfield modules, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Lecture Notes in Math., Vol. 616, Springer, Berlin, 1977, pp. 87–123. MR 0506216
Ronald Linton and Charles Megibben, Extensions of totally projective groups, Proc. Amer. Math. Soc. 64 (1977), no. 1, 35–38. MR 450425, DOI 10.1090/S0002-9939-1977-0450425-0
Charles Megibben, On $p^{\alpha }$-high injectives, Math. Z. 122 (1971), 104–110. MR 299678, DOI 10.1007/BF01110084
Charles Megibben, Totally Zippin $p$-groups, Proc. Amer. Math. Soc. 91 (1984), no. 1, 15–18. MR 735555, DOI 10.1090/S0002-9939-1984-0735555-1
Judy H. Moore, A characterization of Warfield groups, Proc. Amer. Math. Soc. 87 (1983), no. 4, 617–620. MR 687628, DOI 10.1090/S0002-9939-1983-0687628-9
R. B. Warfield Jr., The uniqueness of elongations of Abelian groups, Pacific J. Math. 52 (1974), 289–304. MR 360866
Robert B. Warfield Jr., The structure of mixed abelian groups, Abelian group theory (Proc. Second New Mexico State Univ. Conf., Las Cruces, N.M., 1976) Lecture Notes in Math., Vol. 616, Springer, Berlin, 1977, pp. 1–38. MR 0507073
Robert B. Warfield Jr., Classification theory of abelian groups. II. Local theory, Abelian group theory (Oberwolfach, 1981) Lecture Notes in Math., vol. 874, Springer, Berlin-New York, 1981, pp. 322–349. MR 645938