E-Rings and Related Structures

  • C. Vinsonhaler
Part of the Mathematics and Its Applications book series (MAIA, volume 520)


An E-ring is a ring that is isomorphic to its ring of additive endomorphisms under the left regular representation. That is, a ring R is an E-ring provided REnd(R+) under the map that sends rR to left multiplication by r. An R-module M is called an E-module over R if Hom R (R, M) = Hom Z (R, M). Despite their seemingly specialized definitions, E-rings, E-modules and related notions have played a major role in the theory of torsion-free abelian groups, and pop up with surprising frequency in other subject areas. Here are some examples:
  • [19] A torsion-free abelian group G is cyclic and projective as a module over its endomorphism ring if and only if G = RM, where R is an E-ring and M is an E-module over R.

  • [27] A strongly indecomposable torsion-free abelian group G of finite rank is finitely generated over its endomorphism ring if and only if G is quasi-isomorphic to the additive group of an E-ring.

  • [15] A strongly indecomposable torsion-free group G of finite rank is -uniserial as a module over its endomorphism ring only if G is a local, strongly homogeneous E-ring.

  • [2] A universal algebra A is κ-free if there is a subset X of A of car dinality κ such that every function from X to A extends uniquely to an endomorphism of A. A ring that is l-free as an abelian group is precisely an E-ring (with X = {1}).


Abelian Group Galois Group Galois Extension Endomorphism Ring Finite Rank 
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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  1. [1]
    D. Arnold, R.S. Pierce, J.D. Reid, C. Vinsonhaler and W. Wickless, Torsion-free abelian groups of finite rank projective as modules over their endomorphism rings, J. Algebra 71(1981), 1–10.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    P. Bankston and R. Schutt, On minimally free algebras, Can. J. Math. 37(1985), 963–978.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    R.A. Beaumont and R.S. Pierce, Torsion-free rings, Illinois J. Math. 5(1961), 61–98.MathSciNetzbMATHGoogle Scholar
  4. [4]
    R.A. Beaumont and R.S. Pierce, Subrings of algebraic number fields, Acta. Sci. Math., (Szeged) 22(1961), 202–216.MathSciNetzbMATHGoogle Scholar
  5. [5]
    R.A. Bowshell and P. Schultz, Unital rings whose additive endomorphisms commute, Math. Ann. 228 (1977), 197–214.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    A.L.S. Corner, Every countable reduced torsion-free ring is an endomorphism ring, Proc. London Math. Soc. 13(1963), 687–710.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    A.L.S. Corner and R. Göbel, Prescribing endomorphism algebras, Proc. London Math. Soc. 50(3) (1985), 447–479.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    M. Dugas, Large E-modules exist, J. Algebra 142(1991), 405–413.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    M. Dugas and R. Göbel, Every cotorsion-free ring is an endomorphism ring, Proc. London Math. Soc. 45(1982), 319–336.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [10]
    M. Dugas, A. Mader and C. Vinsonhaler, Large E-rings exist, J. Algebra 108(1987), 88–101.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    T.G. Faticoni, Each countable reduced torsion-free commutative ring is a pure subring of an E-ring, Comm. in Algebra 15(1987), 2545–2564.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    S. Feigelstock, J. Hausen, and R. Raphael, Abelian groups mapping onto their endomorphism rings, preprint.Google Scholar
  13. [13]
    L. Fuchs, Infinite Abelian Groups, Vols. I and II, Academic Press, New York, 1970, 1973.zbMATHGoogle Scholar
  14. [14]
    R. Göbel and S. Shelah, Generalized E-rings, to appear.Google Scholar
  15. [15]
    J. Hausen, Finite rank torsion-free abelian groups uniserial over their endomorphism rings, Proc. Amer. Math. Soc. 93(1985), 227–232.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    J. Hausen, E-transitive torsion-free abelian groups, J. Algebra 107(1987), 17–27.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    J. Hausen and J.A. Johnson, A note on constructing E-rings, Publ. Math. Debrecen 38(1991), 33–38.MathSciNetzbMATHGoogle Scholar
  18. [18]
    A. Mader and C. Vinsonhaler, Torsion-free E-modules, J. Algebra 115(1988), 401–411.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    G.P. Niedzwecki and J.D. Reid, Abelian groups projective over their endomorphism rings, J. Algebra 159 (1993), 139–149.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    R.S. Pierce, Associative Algebras, Graduate Texts in Mathematics 88, Springer-Verlag, New York, 1982.CrossRefGoogle Scholar
  21. [21]
    R.S. Pierce, Subrings of simple algebras, Michigan Math. J. 7(1960), 241–243.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [22]
    R.S. Pierce, E-modules, Abelian Group Theory, Contemporary Mathematics 87 (1989), 221–240.MathSciNetCrossRefGoogle Scholar
  23. [23]
    R.S. Pierce, Realizing Galois fields, Proc. Udine Conf. on Abelian groups and Modules, Springer-Verlag, Vienna (1984), 291–304.Google Scholar
  24. [24]
    R.S. Pierce and C. Vinsonhaler, Realizing algebraic number fields, Abelian Group Theory, Proceedings, Honolulu 1982, Lecture Notes in Math. #1006, Springer-Verlag, Berlin, 1983.Google Scholar
  25. [25]
    R.S. Pierce and C. Vinsonhaler, Classifying E-rings, Comm. in Algebra 19(1991), 615–653.MathSciNetzbMATHCrossRefGoogle Scholar
  26. [26]
    R.S. Pierce and C. Vinsonhaler, Carriers of torsion-free groups, Rend. Sem. Mat. Univ. Padova 84(1990), 263–281.MathSciNetzbMATHGoogle Scholar
  27. [27]
    J.D. Reid, Abelian groups finitely generated over their endomorphism rings, Abelian Group Theory, Lecture Notes in Mathematics 874(1981), 41–52.CrossRefGoogle Scholar
  28. [28]
    P. Schultz, The endomorphism ring of the additive group of a ring, J. Austral. Math. Soc. 15(1973), 60–69.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • C. Vinsonhaler
    • 1
  1. 1.Department of MathematicsUniversity of ConnecticutStorrsUSA

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