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t-Local Domains and Valuation Domains

  • Marco FontanaEmail author
  • Muhammad Zafrullah
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

In a valuation domain (VM), every nonzero finitely generated ideal J is principal and so, in particular, \(J=J^t\); hence, the maximal ideal M is a t-ideal. Therefore, the t-local domains (i.e., the local domains, with maximal ideal being a t-ideal) are “cousins” of valuation domains, but, as we will see in detail, not so close. Indeed, for instance, a localization of a t-local domain is not necessarily t-local, but of course a localization of a valuation domain is a valuation domain. So it is natural to ask under what conditions is a t-local domain a valuation domain? The main purpose of the present paper is to address this question, surveying in part previous work by various authors containing useful properties for applying them to our goal.

Notes

Acknowledgements

The authors would like to thank Francesca Tartarone and Lorenzo Guerrieri for the useful conversations on some aspects of the present paper and the anonymous referee for several helpful suggestions which improved the quality of the manuscript.

References

  1. 1.
    S. Abhyankar, On the valuations centered in a local domain. Amer. J. Math. 78, 321–348 (1956)MathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Abhyankar, Resolution of Singularities of Embedded Algebraic Surfaces (Academic Press, New York, 1966)zbMATHGoogle Scholar
  3. 3.
    D.D. Anderson, G.W. Chang, M. Zafrullah, Integral domains of finite \(t\)-character. J. Algebra 396, 169–183 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    D.F. Anderson, D.E. Dobbs, M. Fontana, On treed nagata rings. J. Pure Appl. Alg. 261, 107–122 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    D.D. Anderson, M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Mat. Ital. A(7), 8 (1994), 397–402Google Scholar
  6. 6.
    D.D. Anderson, M. Zafrullah, The Schreier property and Gauss’ Lemma, Boll. Un. Mat. Ital. 10-B (2007), 43–62Google Scholar
  7. 7.
    E. Bastida, R. Gilmer, Overrings and divisorial idels of rings of the form \(D+M\). Michigan Math. J. 20, 79–95 (1973)MathSciNetCrossRefGoogle Scholar
  8. 8.
    G. Chang, T. Dumitrescu, M. Zafrullah, \(t\)-Splitting sets in integral domains. J. Pure Appl. Algebra 187, 71–86 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    S.U. Chase, Direct product of modules. Trans. Am. Math. Soc. 97, 457–493 (1960)MathSciNetCrossRefGoogle Scholar
  10. 10.
    P.M. Cohn, Bezout rings and their subrings. Proc. Cambridge Phil. Soc. 64, 251–264 (1968)MathSciNetCrossRefGoogle Scholar
  11. 11.
    D. Costa, J. Mott, M. Zafrullah, The construction \(D+XD_{S}[X]\). J. Algebra 53, 423–439 (1978)MathSciNetCrossRefGoogle Scholar
  12. 12.
    E.D. Davis, Overrings of commutative rings, III. Trans. Am. Math. Soc. 182, 175–185 (1973)MathSciNetzbMATHGoogle Scholar
  13. 13.
    D. Dobbs, E. Houston, T. Lucas, M. Zafrullah, \(t\)-linked overrings and PVMDs. Comm. Algebra 17, 2835–2852 (1989)MathSciNetCrossRefGoogle Scholar
  14. 14.
    T. Dumitrescu, Y. Lequain, J. Mott, M. Zafrullah, Almost GCD domains of finite \(t\)-character. J. Algebra 245, 161–181 (2001)MathSciNetCrossRefGoogle Scholar
  15. 15.
    M. Fontana, Topologically defined classes of commutative rings. Ann. Mat. Pura Appl. 123, 331–355 (1980)MathSciNetCrossRefGoogle Scholar
  16. 16.
    M. Fontana, K.A. Loper, Nagata rings, Kronecker function rings, and related semistar operations. Comm. Algebra 31, 4775–4805 (2003)MathSciNetCrossRefGoogle Scholar
  17. 17.
    S. Gabelli, M. Roitman, Maximal divisorial ideals and \(t\)-maximal ideals, JP. J. Algebra Numb. Theor. Appl. 4, 323–336 (2004)zbMATHGoogle Scholar
  18. 18.
    R. Gilmer, Multiplicative Ideal Theory (Marcel Dekker, NewYork, 1972)zbMATHGoogle Scholar
  19. 19.
    R.W. Gilmer, W.J. Heinzer, On the complete integral closure of an integral domain. J. Australian Math. Soc. 6, 351–361 (1966)MathSciNetCrossRefGoogle Scholar
  20. 20.
    R. Gilmer, J. Mott, M. Zafrullah, \(t\)-invertibility and comparability, in Commutative Ring Theory, ed. by P.-J. Cahen, D. Costa, M. Fontana, S.-E. Kabbaj (Marcel Dekker, New York, 1994), pp. 141–150Google Scholar
  21. 21.
    S. Glaz, W.V. Vasconcelos, Flat ideals, II. Manuscripta Math. 22, 325–341 (1977)MathSciNetCrossRefGoogle Scholar
  22. 22.
    M. Griffin, Some results on \(v\)-multiplication rings. Canad. J. Math. 19, 710–722 (1967)MathSciNetCrossRefGoogle Scholar
  23. 23.
    L. Guerrieri, W. Heinzer, B. Olberding, M. Toeniskoetter, Directed unions of local quadratic transforms of regular local rings and pullbacks, in “Rings, Polynomials, and Modules" Fontana, Marco; Frisch, Sophie; Glaz, Sarah; Tartarone, Francesca; Zanardo, Paolo (Editors), Springer International Publishing 2017Google Scholar
  24. 24.
    J.R. Hedstrom, E. Houston, Pseudo-valuation domains. Pacific J. Math. 75, 137–147 (1978)MathSciNetCrossRefGoogle Scholar
  25. 25.
    J.R. Hedstrom, E. Houston, Some remarks on star-operations. J. Pure Appl. Algebra 18, 37–44 (1980)MathSciNetCrossRefGoogle Scholar
  26. 26.
    W. Heinzer, Integral domains in which each non-zero ideal is divisorial. Mathematika 15, 164–170 (1968)MathSciNetCrossRefGoogle Scholar
  27. 27.
    W. Heinzer, K.A. Loper, B. Olberding, H. Schoutens, M. Toeniskoetter, Ideal theory of infinite directed unions of local quadratic transforms. J. Algebra 474, 213–239 (2017)MathSciNetCrossRefGoogle Scholar
  28. 28.
    W. Heinzer, B. Olberding, M.Toeniskoetter, Asymptotic properties of infinite directed unions of local quadratic transforms, J. Algebra 479 (2017), 216–243MathSciNetCrossRefGoogle Scholar
  29. 29.
    W. Heinzer, J. Ohm, An essential ring which is not a \(v\)-multiplication ring. Canad. J. Math. 25, 856–861 (1973)MathSciNetCrossRefGoogle Scholar
  30. 30.
    O. Helmer, Divisibility properties of integral functions. Duke J. Math. 6, 345–356 (1940)MathSciNetCrossRefGoogle Scholar
  31. 31.
    E. Houston, M. Fontana, M.H. Park, Sharpness and semistar operations in Prüfer-like domains, SubmittedGoogle Scholar
  32. 32.
    B.G. Kang, Prüfer \(v\)-multiplication domains and the ring \(R[X]_{N_v}\). J. Algebra 123, 151–170 (1989)Google Scholar
  33. 33.
    I. Kaplansky, Commutative Rings (Allyn and Bacon, 1970)Google Scholar
  34. 34.
    H. Matsumura, Commutative rings (Cambridge University Press, Cambridge, 1986)zbMATHGoogle Scholar
  35. 35.
    S. McAdam, Two conductor theorems. J. Algebra 23, 239–240 (1972)MathSciNetCrossRefGoogle Scholar
  36. 36.
    S. McAdam, D. Rush, Schreier rings. Bull. London Math. Soc. 10, 77–80 (1978)MathSciNetCrossRefGoogle Scholar
  37. 37.
    A. Mimouni, \(TW\)-domains and strong Mori domains. J. Pure Appl. Algebra 177, 79–93 (2003)MathSciNetCrossRefGoogle Scholar
  38. 38.
    A. Mimouni, Integral domains in which each ideal is a \(w\)-ideal. Comm. Algebra 33, 1345–1355 (2005)MathSciNetCrossRefGoogle Scholar
  39. 39.
    J. Mott, M. Zafrullah, On Prüfer \(v\)-multiplication domains. Manuscripta Math. 35, 1–26 (1981)MathSciNetCrossRefGoogle Scholar
  40. 40.
    M. Nagata, Local Rings (Interscience, New York, 1962)zbMATHGoogle Scholar
  41. 41.
    J. Ohm, Some counterexamples related to integral closure in \(D[\![x]\!]\). Trans. Amer. Math. Soc. 122, 321–333 (1966)MathSciNetzbMATHGoogle Scholar
  42. 42.
    G. Picozza, F. Tartarone, When the semistar operation \(\widetilde{\star }\) is the identity. Comm. Algebra. 36, 1954–1975 (2008)MathSciNetCrossRefGoogle Scholar
  43. 43.
    D. Shannon, Monoidal transforms of regular local rings. Amer. J. Math. 95, 294–320 (1973)MathSciNetCrossRefGoogle Scholar
  44. 44.
    F. Wang, \(w\)-modules over a PVMD, In: Proceedings of the International Symposium on Teaching and Applications of Engineering Mathematics, Hong Kong, 2001, pp. 11–120Google Scholar
  45. 45.
    Fanggui Wang, \(w\)-dimension of domains, II. Comm. Algebra 29, 224–28 (2001)MathSciNetCrossRefGoogle Scholar
  46. 46.
    F. Wang, R.L. McCasland, On \(w\)-modules over strong Mori domains, Comm. Algebra 25 (1997), 1285–1306Google Scholar
  47. 47.
    M. Zafrullah, On finite conductor domains. Manuscripta Math. 24, 191–203 (1978)MathSciNetCrossRefGoogle Scholar
  48. 48.
    M. Zafrullah, The \(v\)-operation and intersections of quotient rings of integral domains. Comm. Algebra 13, 1699–1712 (1985)MathSciNetCrossRefGoogle Scholar
  49. 49.
    M. Zafrullah, The \(D+XD_S[X]\) construction from GCD domains. J. Pure Appl. Algebra 50, 93–107 (1988)MathSciNetCrossRefGoogle Scholar
  50. 50.
    M. Zafrullah, On a property of pre-Schreier domains. Comm. Algebra 15, 1895–1920 (1987)MathSciNetCrossRefGoogle Scholar
  51. 51.
    M. Zafrullah, Well behaved prime \(t\)-ideals. J. Pure Appl. Algebra 65, 199–207 (1990)MathSciNetCrossRefGoogle Scholar
  52. 52.
    M. Zafrullah, Putting \(t\)-Invertibility to use, in Non-Noetherian Commutative Ring Theory, ed. by S.T. Chapman, S. Glaz (Kluwer, Dordrecht, 2000), pp. 429–457CrossRefGoogle Scholar

Copyright information

© Springer Nature Singapore Pte Ltd. 2019

Authors and Affiliations

  1. 1.Dipartimento di Matematica e Fisica, Università degli StudiRomeItaly
  2. 2.Department of MathematicsIdaho State UniversityPocatelloUSA

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