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The Divisible and E-Injective Hulls of a Torsion Free Group

  • C. Vinsonhaler
Part of the International Centre for Mechanical Sciences book series (CISM, volume 287)

Abstract

Recently there has been considerable interest in the structure of torsion free abelian groups G as modules over their endomorphism rings E = End(G). In particular, homological properties of the E-module G have been the focus of a number of papers (see (2), (7), (8)). In (2) the structure of finite rank groups G such that G was E-projective was determined, while in (7) the injective hull of G as an E-module was investigated. In the latter paper the question was asked: For which groups G is the E-injective hull of G equal to QG, the divisible hull? This is equivalent to the question: When is QG injective as a QE-module? In this note we obtain a partial answer by imposing an additional condition on G. We examine groups G for which QG is injective as a QE-module and HomQE(QG, QG) = Q(center E). This property, called aEqi, is equivalent to condition weaker than quasi-injectivuty on EG.

Keywords

Abelian Group Endomorphism Ring Finite Rank Torsion Free Group Jacobson Radical 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. (1).
    D. Arnold, Finite Rank Torsion Free Abelian Groups and Rings, Lecture Notes in Math. No. 931, Springer-Verlag, 1982, Berlin-Heidelberg.MATHGoogle Scholar
  2. (2).
    Arnold, Pierce, Reid, Vinsonhaler and Wickless, Torsion free abelian groups of finite rank projective over their endomorphism rings, J. Algebra, Vol. 71, No. 1, (1981), 1–10.CrossRefMATHMathSciNetGoogle Scholar
  3. (3).
    C. Faith, Algebra II, Ring Theory, Springer-Verlag, 1976, Berlin-Heidelberg.CrossRefMATHGoogle Scholar
  4. (4).
    G.D. Poole and J.D. Reid, Abelian groups quasi-injective over their endomorphism rings, Can. J. Math., Vol. 24 No. 4, (1972), 617–621.CrossRefMATHMathSciNetGoogle Scholar
  5. (5).
    J.D. Reid, On the ring of quasi-endomorphisms of a torsion-free abelian group, Topics in Abelian Groups, Scott Foresman, Chicago, 1963.Google Scholar
  6. (6).
    P. Schultz, The endomorphism ring of the additive group of a ring, J. Australian Math. Soc. 15 (1973), 60–69.CrossRefMATHGoogle Scholar
  7. (7).
    C. Vinsonhaler and W. Wickless, Injectivive hulls of torsion free abelian groups as modules over their endomorphism rings, J. Algebra, Vol. 5, No.1,(1979), 64–69.CrossRefMathSciNetGoogle Scholar
  8. (8).
    C. Vinsonhaler and W. Wickless, Torsion free abelian groups quasi-projective over their endomorphism rings, Pac. J. Math. Vol. 68, No. 2, (1977), 527–535.CrossRefMATHMathSciNetGoogle Scholar
  9. (9).
    H. Zassenhaus, Orders as endomorphism rings of modules of the same rank, J. London Math. Soc. 42(1967), 180–182.CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Wien 1984

Authors and Affiliations

  • C. Vinsonhaler
    • 1
  1. 1.University of ConnecticutStorrsUSA

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