Skip to main content
SpringerLink
Log in

Projective Modules, Idempotent Ideals and Intersection Theorems

  • Conference paper
Advances in Ring Theory

Part of the book series: Trends in Mathematics ((TM))

Abstract

We investigate the relationship between projective modules and idempotent ideals for group rings, polynomial rings and more general rings, giving a survey of known results, proving some new results and raising a number of questions. In particular, it is proved that if R is any ring, X a projective right R-module and A an ideal of R such that the R-module X/XA can be generated by a set of elements of cardinality ℵ, for some infinite cardinal ℵ, then X/XB can be generated by a set of elements of cardinality ℵ, where B is the unique maximal idempotent ideal of R contained in A. A recurring theme is that of “intersection theorems” which give information about intersections of powers of ideals of the ring.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. T. Akasaki, Projective R-modules with chain conditions on R/J, Proc. Japan. Acad. 46 (1970), 94–97.

    Article  MATH  MathSciNet  Google Scholar 

  2. F.W. Anderson and K.R. Fuller, Rings and Categories of Modules (Springer-Verlag, New York 1973).

    Google Scholar 

  3. H. Bass, Big projective modules are free, Illinois J. Math. 7 (1963), 24–31.

    MATH  MathSciNet  Google Scholar 

  4. I. Beck, Projective and free modules, Math. Z. 129 (1972), 231–234.

    Article  MATH  MathSciNet  Google Scholar 

  5. G.M. Bergman, Infinite multiplication of ideals in ℵ0-hereditary rings, J. Algebra 24 (1973), 56–70.

    Article  MATH  MathSciNet  Google Scholar 

  6. G. Cauchon, Anneaux semi-premiers noethériens à identités polynômiales, Bull. Soc. Math. France 104 (1976), 99–111.

    MATH  MathSciNet  Google Scholar 

  7. K.W. Gruenberg, The residual nilpotency of certain presentations of finite groups, Archiv. der Math. 13 (1962), 408–417.

    Article  MATH  MathSciNet  Google Scholar 

  8. P. Hall, Finiteness conditions for soluble groups, Proc. London Math. Soc. (3) 4 (1954), 419–436.

    Google Scholar 

  9. I.N. Herstein, A counter-example in Noetherian rings, Proc. Nat. Acad. Sc. 54 (1965), 1036–1037.

    Article  MATH  MathSciNet  Google Scholar 

  10. I.N. Herstein and L.W. Small, The intersection of the powers of the radical in Noetherian P.I. rings, Israel J. Math. 16 (1973), 176–180.

    Article  MathSciNet  Google Scholar 

  11. Y. Hinohara, Projective modules over semilocal rings, Tôhoku Math. J. (2) 14 (1962), 205–211.

    Google Scholar 

  12. Y. Hinohara, Projective modules over weakly noetherian rings, J. Math. Soc. Japan, 15 (1963), 75–88.

    Article  MATH  MathSciNet  Google Scholar 

  13. N. Jacobson, The radical and semisimplicity for arbitrary rings, Amer. J. Math. 67 (1945), 300–320.

    Article  MATH  MathSciNet  Google Scholar 

  14. A.V. Jategaonkar, A counter-example in Ring Theory and Homological Algebra, J. Algebra 12 (1969), 418–440.

    Article  MATH  MathSciNet  Google Scholar 

  15. A.V. Jategaonkar, Jacobson’s conjecture and modules over fully bounded Noetherian rings, J. Algebra 30 (1974), 103–121.

    Article  MATH  MathSciNet  Google Scholar 

  16. S.A. Jennings, The group ring of a class of infinite nilpotent groups, Canad. J. Math. 7 (1955), 169–187.

    Article  MATH  MathSciNet  Google Scholar 

  17. I. Kaplansky,Projective modules, Ann. Math. 68 (1958), 372–377.

    Article  MathSciNet  Google Scholar 

  18. I. Kaplansky, Fields and Rings (University of Chicago Press, Chicago 1969).

    Google Scholar 

  19. I. Kaplansky, Commutative Rings (Allyn and Bacon, Boston 1970).

    MATH  Google Scholar 

  20. G. Krause and T.H. Lenagan, Transfinite powers of the Jacobson radical, Comm. Algebra 7 (1979), 1–8.

    Article  MATH  MathSciNet  Google Scholar 

  21. P.A. Linnell, G. Puninski and P.F. Smith, Idempotent ideals and nonfinitely generated projective modules over integral group rings of polycyclic-by-finite groups, J. Algebra 305 (2006), 845–858.

    Article  MATH  MathSciNet  Google Scholar 

  22. J.C. McConnell. The Intersection Theorem for a class of non-commutative rings, Proc. London Math. Soc. (3) 17 (1967), 487–498.

    Google Scholar 

  23. J.C. McConnell and J.C. Robson, Noncommutative Noetherian Rings (Wiley-Interscience, Chichester 1987).

    MATH  Google Scholar 

  24. W.W. McGovern, G. Puninski and P. Rothmaler, When every projective module is a direct sum of finitely generated modules, J. Algebra 315 (2007), 454–481.

    Article  MATH  MathSciNet  Google Scholar 

  25. Y. Nouazé and P. Gabriel, Ideaux premiers de l’algèbre enveloppante d’une algèbre de Lie nilpotents, J. Algebra 6 (1967), 77–99.

    Article  MATH  MathSciNet  Google Scholar 

  26. M.M. Parmenter and S.K. Sehgal, Idempotent elements and ideals in group rings and the Intersection Theorem, Arch. Math. 24 (1973), 586–600.

    Article  MATH  MathSciNet  Google Scholar 

  27. D.S. Passman, The Algebraic Structure of Group Rings (Wiley-Interscience, New York 1977).

    MATH  Google Scholar 

  28. P. Příhoda, Projective modules are determined by their radical factors, J. Pure Appl. Algebra 210 (2007), 827–835.

    Article  MATH  MathSciNet  Google Scholar 

  29. K.W. Roggenkamp, Integral group rings of solvable finite groups have no idempotent ideals, Archiv Math. (Basel) 25 (1974), 125–128.

    MATH  MathSciNet  Google Scholar 

  30. J.E. Roseblade, The integral group rings of hypercentral groups, Bull. London Math. Soc. 3 (1971), 351–355.

    Article  MATH  MathSciNet  Google Scholar 

  31. J.E. Roseblade, Group rings of polycyclic groups, J. Pure Appl. Algebra 3 (1973), 307–328.

    Article  MATH  MathSciNet  Google Scholar 

  32. J.E. Roseblade and P.F. Smith, A note on hypercentral group rings, J. London Math. Soc. (2) 13 (1976), 13–190.

    Google Scholar 

  33. P.F. Smith, On the Intersection Theorem, Proc. London Math. Soc. (3) 21 (1970), 385–398.

    Google Scholar 

  34. P.F. Smith,Localization and the AR property, Proc. London Math. Soc. (3), 22 (1971), 39–68.

    Google Scholar 

  35. P.F. Smith, Localization in group rings, Proc. London Math. Soc. (3) 22 (1971), 69–90.

    Google Scholar 

  36. P.F. Smith, A note on projective modules, Proc. Roy. Soc. Edin. 75A (1975/76), 23–32.

    Google Scholar 

  37. P.F. Smith, On non-commutative regular local rings, Glasgow Math. J. 17 (1976), 98–102.

    Article  MATH  MathSciNet  Google Scholar 

  38. P.F. Smith, A note on idempotent ideals in group rings, Archiv der Math. 27 (1976), 22–27.

    Article  MATH  Google Scholar 

  39. P.F. Smith, The Artin-Rees property, in Séminaire Dubreil-Malliavin (1981), ed. M.-P. Malliavin, Lecture Notes in Mathematics 924 (Springer-Verlag, Berlin 1981), 197–240.

    Chapter  Google Scholar 

  40. R.G. Swan, The Grothendieck ring of a finite group, Topology 2 (1963), 85–110.

    Article  MATH  MathSciNet  Google Scholar 

  41. R.B. Warfield, Jr., Exchange rings and decompositions of modules, Math. Ann. 199 (1972), 31–36.

    Article  MATH  MathSciNet  Google Scholar 

  42. J.M. Whitehead, Projective modules and their trace ideals, Comm. Algebra 8 (1980), 1873–1901.

    Article  MATH  MathSciNet  Google Scholar 

  43. R. Wisbauer, Foundations of Ring and Module Theory (Gordon and Breach, Philadelphia 1991).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Glasgow, Glasgow, G12 8QW, Scotland, UK

    Patrick F. Smith

Authors
  1. Patrick F. Smith
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Ohio University, 321 Morton Hall, Athens, OH, 45701, USA

    Dinh Van Huynh  & Sergio R. López-Permouth  & 

Additional information

For S.K.Jain on the occasion of his 70th birthday

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Birkhäuser Verlag Basel/Switzerland

About this paper

Cite this paper

Smith, P.F. (2010). Projective Modules, Idempotent Ideals and Intersection Theorems. In: Van Huynh, D., López-Permouth, S.R. (eds) Advances in Ring Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0286-0_20

Download citation

Publish with us

Policies and ethics

Search

Navigation