Skip to main content
SpringerLink
Log in

The behavior of ascending chain conditions on submodules of bounded finite generation in direct sums

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We construct a ring R which has the ascending chain condition on n-generated right ideals for each n ≥ 1 (also called the right pan-acc property), such that no full matrix ring over R has the ascending chain condition on cyclic right ideals. Thus, the right pan-acc property is not a Morita invariant. Moreover, a direct sum of (free) modules with pan-acc does not necessarily even have 1-acc.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. G. M. Bergman, The diamond lemma for ring theory, Advances in Mathematics 29 (1978), 178–218.

    Article  MathSciNet  MATH  Google Scholar 

  2. L. A. Bokut’, Imbeddings into simple associative algebras, Algebra i Logika 15 (1976), 117–142, 245.

    MathSciNet  Google Scholar 

  3. F. Bonang, Noetherian rings whose subidealizer subrings have pan-a.c.c., Communications in Algebra 17 (1989), 1137–1146.

    Article  MathSciNet  MATH  Google Scholar 

  4. E. Sánchez Campos and P. F. Smith, Modules satisfying the ascending chain condition on submodules with bounded uniform dimension, in Rings, Modules and Representations, Contemporary Mathematics, Vol. 480, American Mathematical Society, Providence, RI, 2009, pp. 57–71.

    Chapter  Google Scholar 

  5. P. M. Cohn, Free ideal rings, Journal of Algebra 1 (1964), 47–69.

    Article  MathSciNet  MATH  Google Scholar 

  6. P. M. Cohn, Free Rings and Their Relations, second edition, London Mathematical Society Monographs, Vol. 19, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], London, 1985.

    MATH  Google Scholar 

  7. P. M. Cohn, Free Ideal Rings and Localization in General Rings, New Mathematical Monographs, Vol. 3, Cambridge University Press, Cambridge, 2006.

    Book  MATH  Google Scholar 

  8. D. Frohn, A counterexample concerning ACCP in power series rings, Communications in Algebra 30 (2002), 2961–2966.

    Article  MathSciNet  MATH  Google Scholar 

  9. W. Heinzer and D. Lantz, Commutative rings with ACC on n-generated ideals, Journal of Algebra 80 (1983), 261–278.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. B. Kruskal, The theory of well-quasi-ordering: A frequently discovered concept, Journal of Combinatorial Theory. Series A 13 (1972), 297–305.

    Article  MathSciNet  MATH  Google Scholar 

  11. A.-M. Nicolas, Sur les modules tels que toute suite croissante de sous-modules engendrés par n générateurs soit stationnaire, Journal of Algebra 60 (1979), 249–260.

    Article  MathSciNet  MATH  Google Scholar 

  12. W. Sierpiński, Cardinal and Ordinal Numbers, Polska Akademia Nauk, Monografie Matematyczne, Vol. 34, Państwowe Wydawnictwo Naukowe, Warsaw, 1958.

    MATH  Google Scholar 

  13. M. E. Antunes Sim˜oes and P. F. Smith, Rings whose free modules satisfy the ascending chain condition on submodules with a bounded number of generators, Journal of Pure and Applied Algebra 123 (1998), 51–66.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Pace P. Nielsen

Authors
  1. Pace P. Nielsen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pace P. Nielsen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nielsen, P.P. The behavior of ascending chain conditions on submodules of bounded finite generation in direct sums. Isr. J. Math. 215, 339–347 (2016). https://doi.org/10.1007/s11856-016-1381-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-016-1381-y

Search

Navigation