Abstract
Several equivalent descriptions are given of the class of primary abelian groups whose separable subgroups are all direct sums of cyclic groups; such groups are called ω-totally Σ-cyclic. This establishes the converse of a theorem due to Megibben. For n < ω, this is generalized to a consideration of the class of primary abelian groups whose p ω+n-bounded subgroups are all p ω+n-projective. The question of whether there are such groups that are proper in the sense that they are neither p ω+n-projective nor ω-totally Σ-cyclic is shown to be logically equivalent to a natural question about the structure of valuated vector spaces. Finally, it is shown that both of these statements are independent of ZFC.
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The authors would like to thank the referee for his or her many useful suggestions, as well as to thank the editor, Professor Francesco Altomare, for his care in processing our manuscript.
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Danchev, P.V., Keef, P.W. An Application of Set Theory to ω + n-Totally p ω+n-Projective Primary Abelian Groups. Mediterr. J. Math. 8, 525–542 (2011). https://doi.org/10.1007/s00009-010-0088-2
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DOI: https://doi.org/10.1007/s00009-010-0088-2