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t-Local Domains and Valuation Domains

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Advances in Commutative Algebra

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

Abstract

In a valuation domain (V, M), 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.

Dedicated to David F. Anderson

The first named author was partially supported by GNSAGA of Istituto Nazionale di Alta Matematica.

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References

  1. S. Abhyankar, On the valuations centered in a local domain. Amer. J. Math. 78, 321–348 (1956)

    Article  MathSciNet  Google Scholar 

  2. S. Abhyankar, Resolution of Singularities of Embedded Algebraic Surfaces (Academic Press, New York, 1966)

    MATH  Google Scholar 

  3. D.D. Anderson, G.W. Chang, M. Zafrullah, Integral domains of finite \(t\)-character. J. Algebra 396, 169–183 (2013)

    Article  MathSciNet  Google Scholar 

  4. D.F. Anderson, D.E. Dobbs, M. Fontana, On treed nagata rings. J. Pure Appl. Alg. 261, 107–122 (1989)

    Article  MathSciNet  Google Scholar 

  5. D.D. Anderson, M. Zafrullah, On a theorem of Kaplansky, Boll. Un. Mat. Ital. A(7), 8 (1994), 397–402

    Google Scholar 

  6. D.D. Anderson, M. Zafrullah, The Schreier property and Gauss’ Lemma, Boll. Un. Mat. Ital. 10-B (2007), 43–62

    Google Scholar 

  7. E. Bastida, R. Gilmer, Overrings and divisorial idels of rings of the form \(D+M\). Michigan Math. J. 20, 79–95 (1973)

    Article  MathSciNet  Google Scholar 

  8. G. Chang, T. Dumitrescu, M. Zafrullah, \(t\)-Splitting sets in integral domains. J. Pure Appl. Algebra 187, 71–86 (2004)

    Article  MathSciNet  Google Scholar 

  9. S.U. Chase, Direct product of modules. Trans. Am. Math. Soc. 97, 457–493 (1960)

    Article  MathSciNet  Google Scholar 

  10. P.M. Cohn, Bezout rings and their subrings. Proc. Cambridge Phil. Soc. 64, 251–264 (1968)

    Article  MathSciNet  Google Scholar 

  11. D. Costa, J. Mott, M. Zafrullah, The construction \(D+XD_{S}[X]\). J. Algebra 53, 423–439 (1978)

    Article  MathSciNet  Google Scholar 

  12. E.D. Davis, Overrings of commutative rings, III. Trans. Am. Math. Soc. 182, 175–185 (1973)

    MathSciNet  MATH  Google Scholar 

  13. D. Dobbs, E. Houston, T. Lucas, M. Zafrullah, \(t\)-linked overrings and PVMDs. Comm. Algebra 17, 2835–2852 (1989)

    Article  MathSciNet  Google Scholar 

  14. T. Dumitrescu, Y. Lequain, J. Mott, M. Zafrullah, Almost GCD domains of finite \(t\)-character. J. Algebra 245, 161–181 (2001)

    Article  MathSciNet  Google Scholar 

  15. M. Fontana, Topologically defined classes of commutative rings. Ann. Mat. Pura Appl. 123, 331–355 (1980)

    Article  MathSciNet  Google Scholar 

  16. M. Fontana, K.A. Loper, Nagata rings, Kronecker function rings, and related semistar operations. Comm. Algebra 31, 4775–4805 (2003)

    Article  MathSciNet  Google Scholar 

  17. S. Gabelli, M. Roitman, Maximal divisorial ideals and \(t\)-maximal ideals, JP. J. Algebra Numb. Theor. Appl. 4, 323–336 (2004)

    MATH  Google Scholar 

  18. R. Gilmer, Multiplicative Ideal Theory (Marcel Dekker, NewYork, 1972)

    MATH  Google Scholar 

  19. R.W. Gilmer, W.J. Heinzer, On the complete integral closure of an integral domain. J. Australian Math. Soc. 6, 351–361 (1966)

    Article  MathSciNet  Google Scholar 

  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. 141–150

    Google Scholar 

  21. S. Glaz, W.V. Vasconcelos, Flat ideals, II. Manuscripta Math. 22, 325–341 (1977)

    Article  MathSciNet  Google Scholar 

  22. M. Griffin, Some results on \(v\)-multiplication rings. Canad. J. Math. 19, 710–722 (1967)

    Article  MathSciNet  Google Scholar 

  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 2017

    Google Scholar 

  24. J.R. Hedstrom, E. Houston, Pseudo-valuation domains. Pacific J. Math. 75, 137–147 (1978)

    Article  MathSciNet  Google Scholar 

  25. J.R. Hedstrom, E. Houston, Some remarks on star-operations. J. Pure Appl. Algebra 18, 37–44 (1980)

    Article  MathSciNet  Google Scholar 

  26. W. Heinzer, Integral domains in which each non-zero ideal is divisorial. Mathematika 15, 164–170 (1968)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  28. W. Heinzer, B. Olberding, M.Toeniskoetter, Asymptotic properties of infinite directed unions of local quadratic transforms, J. Algebra 479 (2017), 216–243

    Article  MathSciNet  Google Scholar 

  29. W. Heinzer, J. Ohm, An essential ring which is not a \(v\)-multiplication ring. Canad. J. Math. 25, 856–861 (1973)

    Article  MathSciNet  Google Scholar 

  30. O. Helmer, Divisibility properties of integral functions. Duke J. Math. 6, 345–356 (1940)

    Article  MathSciNet  Google Scholar 

  31. E. Houston, M. Fontana, M.H. Park, Sharpness and semistar operations in Prüfer-like domains, Submitted

    Google Scholar 

  32. B.G. Kang, Prüfer \(v\)-multiplication domains and the ring \(R[X]_{N_v}\). J. Algebra 123, 151–170 (1989)

    Google Scholar 

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

    Google Scholar 

  34. H. Matsumura, Commutative rings (Cambridge University Press, Cambridge, 1986)

    MATH  Google Scholar 

  35. S. McAdam, Two conductor theorems. J. Algebra 23, 239–240 (1972)

    Article  MathSciNet  Google Scholar 

  36. S. McAdam, D. Rush, Schreier rings. Bull. London Math. Soc. 10, 77–80 (1978)

    Article  MathSciNet  Google Scholar 

  37. A. Mimouni, \(TW\)-domains and strong Mori domains. J. Pure Appl. Algebra 177, 79–93 (2003)

    Article  MathSciNet  Google Scholar 

  38. A. Mimouni, Integral domains in which each ideal is a \(w\)-ideal. Comm. Algebra 33, 1345–1355 (2005)

    Article  MathSciNet  Google Scholar 

  39. J. Mott, M. Zafrullah, On Prüfer \(v\)-multiplication domains. Manuscripta Math. 35, 1–26 (1981)

    Article  MathSciNet  Google Scholar 

  40. M. Nagata, Local Rings (Interscience, New York, 1962)

    MATH  Google Scholar 

  41. J. Ohm, Some counterexamples related to integral closure in \(D[\![x]\!]\). Trans. Amer. Math. Soc. 122, 321–333 (1966)

    MathSciNet  MATH  Google Scholar 

  42. G. Picozza, F. Tartarone, When the semistar operation \(\widetilde{\star }\) is the identity. Comm. Algebra. 36, 1954–1975 (2008)

    Article  MathSciNet  Google Scholar 

  43. D. Shannon, Monoidal transforms of regular local rings. Amer. J. Math. 95, 294–320 (1973)

    Article  MathSciNet  Google Scholar 

  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. 11–120

    Google Scholar 

  45. Fanggui Wang, \(w\)-dimension of domains, II. Comm. Algebra 29, 224–28 (2001)

    Article  MathSciNet  Google Scholar 

  46. F. Wang, R.L. McCasland, On \(w\)-modules over strong Mori domains, Comm. Algebra 25 (1997), 1285–1306

    Google Scholar 

  47. M. Zafrullah, On finite conductor domains. Manuscripta Math. 24, 191–203 (1978)

    Article  MathSciNet  Google Scholar 

  48. M. Zafrullah, The \(v\)-operation and intersections of quotient rings of integral domains. Comm. Algebra 13, 1699–1712 (1985)

    Article  MathSciNet  Google Scholar 

  49. M. Zafrullah, The \(D+XD_S[X]\) construction from GCD domains. J. Pure Appl. Algebra 50, 93–107 (1988)

    Article  MathSciNet  Google Scholar 

  50. M. Zafrullah, On a property of pre-Schreier domains. Comm. Algebra 15, 1895–1920 (1987)

    Article  MathSciNet  Google Scholar 

  51. M. Zafrullah, Well behaved prime \(t\)-ideals. J. Pure Appl. Algebra 65, 199–207 (1990)

    Article  MathSciNet  Google Scholar 

  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. 429–457

    Chapter  Google Scholar 

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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.

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Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università degli Studi, “Roma Tre”, 00146, Rome, Italy

    Marco Fontana

  2. Department of Mathematics, Idaho State University, Pocatello, ID, 83209-8085, USA

    Muhammad Zafrullah

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  1. Marco Fontana
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  2. Muhammad Zafrullah
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Correspondence to Marco Fontana .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, American University of Sharjah, Sharjah, United Arab Emirates

    Ayman Badawi

  2. Department of Mathematical Sciences, Clemson University, Clemson, SC, USA

    Jim Coykendall

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Fontana, M., Zafrullah, M. (2019). t-Local Domains and Valuation Domains. In: Badawi, A., Coykendall, J. (eds) Advances in Commutative Algebra. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-7028-1_3

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