On Type-Related Properties of Torsionfree Abelian Groups
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.
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