Extensions of Simple Modules and the Converse of Schur’s Lemma

  • Greg Marks
  • Markus Schmidmeier
Part of the Trends in Mathematics book series (TM)


The converse of Schur’s lemma (or CSL) condition on a module category has been the subject of considerable study in recent years. In this note we extend that work by developing basic properties of module categories in which the CSL condition governs modules of finite length.

Mathematics Subject Classification (2000)

Primary 16D90 16G216S50 


Converse of Schur’s Lemma Gabriel quiver 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Alaoui, A. Haily, Perfect rings for which the converse of Schur’s lemma holds, Publ. Mat. 45 (2001), no. 1, 219–222.zbMATHMathSciNetGoogle Scholar
  2. [2]
    M. Alaoui, A. Haily, The converse of Schur’s lemma in noetherian rings and group algebras, Comm. Algebra 33 (2005), no. 7, 2109–2114.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    V.V. Bavula, The extension group of the simple modules over the first Weyl algebra, Bull. London Math. Soc. 32 (2000), no. 2, 182–190.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J.H. Cozzens, Homological properties of the ring of differential polynomials, Bull. Amer. Math. Soc. 76 (1970), 75–79.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    M. Dombrovskaya, G. Marks, Asymmetry in the converse of Schur’s Lemma, Comm. Algebra, to appear.Google Scholar
  6. [6]
    D. Eisenbud, J.C. Robson, Hereditary Noetherian prime rings, J. Algebra 16 (1970), 86–104.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    C. Faith, Indecomposable injective modules and a theorem of Kaplansky, Comm. Algebra 30 (2002), no. 12, 5875–5889.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    K.R. Goodearl, R.B. Warfield Jr., Simple modules over hereditary Noetherian prime rings, J. Algebra 57 (1979), no. 1, 82–100.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    Y. Hirano, J.K. Park, Rings for which the converse of Schur’s Lemma holds, Math. J. Okayama Univ. 33 (1991), 121–131.zbMATHMathSciNetGoogle Scholar
  10. [10]
    C. Huh, C.O. Kim, π-regular rings satisfying the converse of Schur’s lemma, Math. J. Okayama Univ. 34 (1992), 153–156.zbMATHMathSciNetGoogle Scholar
  11. [11]
    J.C. McConnell, J.C. Robson, Homomorphisms and extensions of modules over certain differential polynomial rings, J. Algebra 26 (1973), 319–342.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    J.C. Robson, Non-commutative Dedekind rings, J. Algebra 9 (1968), 249–265.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    M. Schmidmeier, A family of noetherian rings with their finite length modules under control, Czechoslovak Math. J. 52(127) (2002), no. 3, 545–552.CrossRefMathSciNetGoogle Scholar
  14. [14]
    R. Ware, J. Zelmanowitz, Simple endomorphism rings, Amer. Math. Monthly 77 (1970), no. 9, 987–989.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Greg Marks
    • 1
  • Markus Schmidmeier
    • 2
  1. 1.Department of Mathematics and Computer ScienceSt. Louis UniversitySt. LouisUSA
  2. 2.Mathematical SciencesFlorida Atlantic UniversityBoca RatonUSA

Personalised recommendations