Vol. 79, No. 2, 1978

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Characterization of a class of torsion free groups in terms of endomorphisms

Eugene Frank Cornelius, Jr.

Vol. 79 (1978), No. 2, 341–355

Characterizalions in terms of endomorphisms and quasi-endomorphisms are obtained for torsion free abelian groups with the property that each pure subgroup of finite rank is a quasi-summand. A group has this property if and only if its ring of endomorphisms with finite rank is 2-fold ct-transitive, and hence k-fold ct-transitive for every k. This property is equivalent to complete decomposability for countable groups the type set of which satisfies the maximum condition. A stronger version of transitivity is required to describe separable groups the type set of which satisfies the maximum condition; to insure generality, it is shown that the maximum condition does not imply countability of the type set, a result of independent interest.

Mathematical Subject Classification 2000
Primary: 20K20
Received: 7 June 1978
Published: 1 December 1978
Eugene Frank Cornelius, Jr.