Periodica Mathematica Hungarica

, Volume 69, Issue 1, pp 21–31 | Cite as

The Hopfian exponent of an abelian group

  • Brendan GoldsmithEmail author
  • Peter Vámos


If \(G\) is a Hopfian abelian group then it is, in general, difficult to determine if direct sums of copies of \(G\) will remain Hopfian. We exhibit large classes of Hopfian groups such that every finite direct sum of copies of the group is Hopfian. We also show that for any integer \(n > 1\) there is a torsion-free Hopfian group \(G\) having the property that the direct sum of \(n\) copies of \(G\) is not Hopfian but the direct sum of any lesser number of copies is Hopfian.


Abelian groups Hopfian groups Leavitt rings 

Mathematics Subject Classification

Primary 20K30 Secondary 20K20 20K10 


  1. 1.
    R.A. Beaumont, R.S. Pierce, Isomorphic direct summands of abelian groups. Math. Ann. 153, 21–37 (1964)Google Scholar
  2. 2.
    P.M. Cohn, Some remarks on the invariant basis property. Topology 5, 215–228 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    A.L.S. Corner, Every countable reduced torsion-free ring is an endomorphism ring. Proc. Lond. Math. Soc. 13, 687–710 (1963)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    A.L.S. Corner, Three examples on hopficity in torsion-free abelian groups. Acta Math. Acad. Sci. Hungar. 16, 303–310 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    A.L.S. Corner, R. Göbel, Prescribibg endomorphism algebras: a unified treatment. Proc. Lond. Math. Soc. 50, 447–479 (1985)CrossRefzbMATHGoogle Scholar
  6. 6.
    L. Fuchs, Infinite Abelian Groups, vol. I (Academic Press, New York, 1970)zbMATHGoogle Scholar
  7. 7.
    L. Fuchs, Infinite Abelian Groups, vol. II (Academic Press, New York, 1973)zbMATHGoogle Scholar
  8. 8.
    B. Goldsmith, Anthony Leonard Southern Corner 1934–2006, in Models, Modules and Abelian Groups, ed. by R. Göbel, B. Goldsmith (Walter de Gruyter, Berlin, 2008), pp. 1–7Google Scholar
  9. 9.
    B. Goldsmith, K. Gong, A note on hopfian and co-hopfian abelian groups. Contemp. Math. 576, 129–136 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    B. Goldsmith, L. Strüngmann, Torsion-free weakly transitive abelian group. Comm. Algebra 33, 1171–1191 (2005)MathSciNetCrossRefGoogle Scholar
  11. 11.
    K.R. Goodearl, von Neumann Regular Rings (Pitman (Advanced Publishing Program), Boston, 1979)zbMATHGoogle Scholar
  12. 12.
    K.R. Goodearl, Surjective endomorphisms of finitely generated modules. Comm. Algebra 15, 589–609 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    I. Kaplansky, Modules over Dedekind rings and valuation rings. Trans. Am. Math. Soc. 72, 327–340 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    I. Kaplansky, Infinite Abelian Groups (University of Michigan Press, Ann Arbour, 1954 and 1969)Google Scholar
  15. 15.
    J.C. McConnell, J.C. Robson, Noncommutative Noetherian Rings, revised ed., Graduate Studies in Mathematics, vol. 30 (American Mathematical Society, Providence, RI, 2001) (With the cooperation of L. W. Small)Google Scholar
  16. 16.
    N.H. McCoy, Divisors of zero in matric rings. Bull. Am. Math. Soc 47, 166–172 (1941)MathSciNetCrossRefGoogle Scholar
  17. 17.
    P.M. Neumann, Pathology in the Representation Theory of Infinite Soluble Groups, in Proceedings of ‘ Groups-Korea 1988’. Lecture Notes in Mathematics 1398 (Eds A.C. Kim and B.H. Neumann), pp. 124–139Google Scholar
  18. 18.
    R.S. Pierce, Homomorphisms of Primary Abelian Groups, Topics in Abelian Groups (Scott Foresman, Chicago, 1963), pp. 215–310Google Scholar

Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesDublin Institute of TechnologyDublin 8Ireland
  2. 2.Department of MathematicsUniversity of ExeterExeterUK

Personalised recommendations