Variations of purity in abelian group theory

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In a collection of classical papers, R. Nunke studied radicals R on the category of abelian groups constructed using extensions of the integers. In particular, the sequences in RExt AG were called R-pure. In this paper several types of purity related to R-purity are discussed. Projective and injective resolutions of an arbitrary group are constructed, providing interesting examples of dual classes of groups. One particular variation, that of R ∗ -purity, is shown to have many of the advantages of R-purity, while suffering from fewer of its drawbacks, e.g., every group has an R ∗ -injective hull and the class of R ∗ -projectives is closed under arbitrary subgroups. The most important example is R= p α , where α is an ordinal. When α is a limit, p α -purity and its generalizations are closely related to the completion of groups in the α-topology. Although these are the motivating examples, many of the results are stated in a substantially more general context.