Variations of purity in abelian group theory

[摘要] 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(A, G) 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(alpha), where alpha is an ordinal. When alpha is a limit, p(alpha)-purity and its generalizations are closely related to the completion of groups in the alpha-topology. Although these are the motivating examples, many of the results are stated in a substantially more general context. (C) 1999 Elsevier Science B.V. All rights reserved.

[发布日期] 1999-12-30