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On Type-Related Properties of Torsionfree Abelian Groups

  • Claudia Metelli
Part of the Lecture Notes in Mathematics book series (LNM, volume 1006)

Abstract

This Note investigates certain properties of torsion--free abelian groups which are related to types. We start by intro-ducing for every type t two functorial subgroups of G called G[t], G *[t], which are in some way duals to the classical G(t), G *(t). Paragraph 1 is dedicated to studying the general properties of the new subgroups, their relations to the old ones, and those properties of G which naturally follow from these relations. For instance, we are lead to the source, in G, of rank 1, type t summands of G. In para- graph 2, by slightly strenghthening two of the general properties obtained, we get a class c of t.f. groups which, besides having some nice closure properties and containing separable groups and vector groups, is defined “locally at the type t”: i.e. to see that G E c one studies the behaviour of G w.r. to type t “one type at a time”. Finally, in Paragraph 3, by introducing and investigating separability as an element property rather than a group property, and by using “localization at the type t”, we get as a bonus a very simple proof of the fundamental result by Fuchs stating that summands of separable groups are separable.

Keywords

Direct Summand Finite Rank Free Abelian Group Separable Group Vector Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. [C]
    E.F.Cornelius, Jr., A sufficient condition for separability J.Alg. 67 (2), (1980), 476–478.Google Scholar
  2. [F II]
    F II]L.Fuchs, Infinite Abelian Groups Vol. II, London-New York: Academic Press (1974).Google Scholar
  3. [M S]
    C.Metelli - L.Salce, The endomorphism ring of an abelian torsion--free homogeneous separable group Arch.Mat., 26 (1975), 480–485.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Claudia Metelli

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