Vol. 29, No. 2, 1969

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A note on a theorem of Hill

Phillip Alan Griffith

Vol. 29 (1969), No. 2, 279–284

Recently Hill has shown the existence of an abelian p-group with the property that each infinite subgroup can be embedded in a direct summand of the same cardinality but the group is not a direct sum of countable groups. Megibben has since observed that this phenomenon occurs in a larger class of abelian groups. In this note we show that such pathology is present in modules for a rather wide class of rings. In fact, the lack of such phenomena for a particular class of modules serves as a characterization for left perfect rings. Our results also yield some facts concerning pure injective modules.

Mathematical Subject Classification
Primary: 16.40
Received: 29 April 1968
Published: 1 May 1969
Phillip Alan Griffith
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 W. Green Street
Urbana IL 61801-2975
United States