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Israel Journal of Mathematics

, Volume 215, Issue 1, pp 339–347 | Cite as

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

  • Pace P. Nielsen
Article

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.

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References

  1. [1]
    G. M. Bergman, The diamond lemma for ring theory, Advances in Mathematics 29 (1978), 178–218.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    L. A. Bokut’, Imbeddings into simple associative algebras, Algebra i Logika 15 (1976), 117–142, 245.MathSciNetGoogle Scholar
  3. [3]
    F. Bonang, Noetherian rings whose subidealizer subrings have pan-a.c.c., Communications in Algebra 17 (1989), 1137–1146.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [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.CrossRefGoogle Scholar
  5. [5]
    P. M. Cohn, Free ideal rings, Journal of Algebra 1 (1964), 47–69.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [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.zbMATHGoogle Scholar
  7. [7]
    P. M. Cohn, Free Ideal Rings and Localization in General Rings, New Mathematical Monographs, Vol. 3, Cambridge University Press, Cambridge, 2006.CrossRefzbMATHGoogle Scholar
  8. [8]
    D. Frohn, A counterexample concerning ACCP in power series rings, Communications in Algebra 30 (2002), 2961–2966.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    W. Heinzer and D. Lantz, Commutative rings with ACC on n-generated ideals, Journal of Algebra 80 (1983), 261–278.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    J. B. Kruskal, The theory of well-quasi-ordering: A frequently discovered concept, Journal of Combinatorial Theory. Series A 13 (1972), 297–305.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [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.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    W. Sierpiński, Cardinal and Ordinal Numbers, Polska Akademia Nauk, Monografie Matematyczne, Vol. 34, Państwowe Wydawnictwo Naukowe, Warsaw, 1958.zbMATHGoogle Scholar
  13. [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.MathSciNetCrossRefzbMATHGoogle Scholar

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© Hebrew University of Jerusalem 2016

Authors and Affiliations

  1. 1.Department of MathematicsBrigham Young UniversityProvoUSA

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