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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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$Qqpi$ groups and quasi-equivalence
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by H. P. Goeters and W. J. Wickless PDF
Proc. Amer. Math. Soc. 126 (1998), 3145-3150 Request permission


A torsion-free abelian group $G$ is $qpi$ if every map from a pure subgroup $K$ of $G$ into $G$ lifts to an endomorphism of $G.$ The class of $qpi$ groups has been extensively studied, resulting in a number of nice characterizations. We obtain some characterizations for the class of homogeneous $Qqpi$ groups, those homogeneous groups $G$ such that, for $K$ pure in $G,$ every $\theta :K\rightarrow G$ has a lifting to a quasi-endomorphism of $G.$ An irreducible group is $Qqpi$ if and only if every pure subgroup of each of its strongly indecomposable quasi-summands is strongly indecomposable. A $Qqpi$ group $G$ is $qpi$ if and only if every endomorphism of $G$ is an integral multiple of an automorphism. A group $G$ has minimal test for quasi-equivalence ($mtqe)$ if whenever $K$ and $L$ are quasi-isomorphic pure subgroups of $G$ then $K$ and $L$ are equivalent via a quasi-automorphism of $G.$ For homogeneous groups, we show that in almost all cases the $Qqpi$ and $mtqe$ properties coincide.
  • David M. Arnold, Finite rank torsion free abelian groups and rings, Lecture Notes in Mathematics, vol. 931, Springer-Verlag, Berlin-New York, 1982. MR 665251, DOI 10.1007/BFb0094245
  • D. M. Arnold, B. Oโ€™Brien, and J. D. Reid, Quasipure injective and projective torsion-free abelian groups of finite rank, Proc. London Math. Soc. (3) 38 (1979), no.ย 3, 532โ€“544. MR 532986, DOI 10.1112/plms/s3-38.3.532
  • James D. Reid, On the ring of quasi-endomorphisms of a torsion-free group, Topics in Abelian Groups (Proc. Sympos., New Mexico State Univ., 1962), Scott, Foresman & Co., Chicago, Ill., 1963, pp.ย 51โ€“68. MR 0169915
  • J. D. Reid, Abelian groups cyclic over their endomorphism rings, Abelian group theory (Honolulu, Hawaii, 1983) Lecture Notes in Math., vol. 1006, Springer, Berlin, 1983, pp.ย 190โ€“203. MR 722618, DOI 10.1007/BFb0103702
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Additional Information
  • H. P. Goeters
  • Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849
  • Email:
  • W. J. Wickless
  • Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
  • Email:
  • Received by editor(s): March 21, 1997
  • Communicated by: Ronald M. Solomon
  • © Copyright 1998 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 126 (1998), 3145-3150
  • MSC (1991): Primary 20K15
  • DOI:
  • MathSciNet review: 1485477