groups and quasi-equivalence

Authors:
H. P. Goeters and W. J. Wickless

Journal:
Proc. Amer. Math. Soc. **126** (1998), 3145-3150

MSC (1991):
Primary 20K15

MathSciNet review:
1485477

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Abstract: A torsion-free abelian group is if every map from a pure subgroup of into lifts to an endomorphism of The class of groups has been extensively studied, resulting in a number of nice characterizations. We obtain some characterizations for the class of homogeneous groups, those homogeneous groups such that, for pure in every has a lifting to a quasi-endomorphism of An irreducible group is if and only if every pure subgroup of each of its strongly indecomposable quasi-summands is strongly indecomposable. A group is if and only if every endomorphism of is an integral multiple of an automorphism. A group has minimal test for quasi-equivalence ( if whenever and are quasi-isomorphic pure subgroups of then and are equivalent via a quasi-automorphism of For homogeneous groups, we show that in almost all cases the and properties coincide.

**[A]**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****[A-O'B-R]**D. M. Arnold, B. O’Brien, and J. D. Reid,*Quasipure injective and projective torsion-free abelian groups of finite rank*, Proc. London Math. Soc. (3)**38**(1979), no. 3, 532–544. MR**532986**, 10.1112/plms/s3-38.3.532**[R-1]**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 and Co., Chicago, Ill., 1963, pp. 51–68. MR**0169915****[R-2]**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. 190–203. MR**722618**, 10.1007/BFb0103702

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Additional Information

**H. P. Goeters**

Affiliation:
Department of Mathematics, Auburn University, Auburn, Alabama 36849

Email:
goetehp@mail.auburn.edu

**W. J. Wickless**

Affiliation:
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Email:
wjwick@uconnvm.uconn.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-98-04734-0

Received by editor(s):
March 21, 1997

Communicated by:
Ronald M. Solomon

Article copyright:
© Copyright 1998
American Mathematical Society