A-Projective Groups of Large Cardinality

  • Ulrich Albrecht
Conference paper
Part of the International Centre for Mechanical Sciences book series (CISM, volume 287)


Because free abelian groups have many useful homological properties, the question arises how closely do groups of the form ⊕ A for a torsion-free group A resemble free groups. It becomes apparent that it is necessary to restrict the choices for A. In [1] and [2], it has been shown that the best way of doing this is to restrict the class of rings from which the endomorphism ring E(A) = Horn (A, A) can be chosen. In these papers, A was a torsion-free, reduced abelian group whose endomorphism ring is semi-prime, right and left Noetherian, and hereditary. Such groups will be called generalized rank 1 groups. Their important properties are given in [2, Proposition 3.2]. These can be used to show that the A-projective groups which are direct summands of groups of the form ⊕I A closely resemble free abelian groups.


Abelian Group Direct Summand Generalize Rank Endomorphism Ring Cardinal Number 
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  1. 1.
    Albrecht, U.; Endomorphism rings and A-projective torsion-free abelian groups, in Abelian Group Theory, Proceedings Honolulu 1982/83, Springer LNM 1006, New York, 1983, 209–227.Google Scholar
  2. 2.
    Albrecht, U.; A note on locally A-projective abelian groups, to appear.Google Scholar
  3. 3.
    Chase, S.; On group extensions and a problem of J. H. C. Whitehead, in Topics in Abelian Groups, Scott — Foresman, Glenview, 1963, 173–193.Google Scholar
  4. 4.
    Eklof, P.; On the existence of K-free abelian groups, Proc. Amer. Math. Soc. 47, 1975, 65–72.MATHMathSciNetGoogle Scholar
  5. 5.
    Eklof, P. and Huber, M.; Abelian group extensions and the axiom of constructibility, Comment. Math. Helvetii, 54, 1979, 440 – 457.CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    Kaplansky, I.; Projective Modules, Annals of Mathematics, 68, No. 2, 1958, 372 – 376.CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Shelah, S.; A compactness theorem for singular cardinals, free algebras, Whitehead problem and transversals, Israel G. Math., 21, 1975, 319 – 349.CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Wien 1984

Authors and Affiliations

  • Ulrich Albrecht
    • 1
  1. 1.Marshall UniversityUSA

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