Putting T-Invertibility to Use

  • Muhammad Zafrullah
Part of the Mathematics and Its Applications book series (MAIA, volume 520)


This article gives a survey of how the notion of t-invertibility has, in recent years, been used to develop new concepts that enhance our understanding of the multiplicative structure of commutative integral domains. The concept of t-invertibility arises in the context of star operations. However, in general terms a (fractional) ideal A, of an integral domain D, is t-invertible if there is a finitely generated (fractional) ideal FA and a finitely generated fractional ideal G A -l such that (FG)-1 = D. In a more specialized context the notion of t-invertibility has to do with the t-operation which is one of the so called star operations. There seems to be no book other than Gilmer’s [Gil] that treats star operations purely from a ring theoretic view point. But a lot has changed since Gilmer’s book was published. So I have devoted a part of section 1. to an introduction to star operations, *-invertibility in general, and t-invertibility in particular.


Prime Ideal Integral Domain Integral Ideal Principal Ideal Nonzero Ideal 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [And]
    D.D. Anderson, “Star operations induced by overrings” Comm. Algebra 16(1988) 2535–2553.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [A 1]
    D.D. Anderson, “GCD domains, ‘Gauss’ Lemma, and contents of polynomials” (thisGoogle Scholar
  3. collection).Google Scholar
  4. [AA]
    D.D. Anderson and D.F. Anderson, “Generalized GCD domains” Comment. Math. Univ. St. Pauli 28(1979) 215–221.Google Scholar
  5. [AA 1]
    D.D. Anderson and D.F. Anderson, “Examples of star operations on integral domains” Comm. Algebra, 15(5)(1990) 1621–1643.CrossRefGoogle Scholar
  6. [AAZ]
    D.D. Anderson, D.F. Anderson and M. Zafrullah, “Splitting the t-class group” J. Pure Appl. Algebra 74(1991) 17–37.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [AAZ 1]
    D.D. Anderson, D.F. Anderson and M. Zafrullah, “Rings between D[X] and K[X]” Houston J. Math. 17(1991), 109–129.MathSciNetzbMATHGoogle Scholar
  8. [AAZ 2]
    D.D. Anderson, D.F. Anderson and M. Zafrullah, “On generalized unique factorization” Bollettino U. M. I. (7) 9-A (1995), 401–413MathSciNetGoogle Scholar
  9. [AC]
    D.D. Anderson and S. Cook,“Two star operations and their induced lattice”, Comm. Algebra to appear.Google Scholar
  10. [AHZ]
    D.D. Anderson, E. Houston and M. Zafrullah, “t-linked extensions, the t-class group and Nagata’s Theorem” J. Pure Appl. Algebra 86(1993) 109–124.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [AM]
    D.D. Anderson and L. Mahaney, “On primary factorization” J. Pure Appl. Algebra 54(1988) 141–154.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [AMZ 1]
    D.D. Anderson, J. Mott and M. Zafrullah, “Some quotient based statements in multiplicative ideal theory” Bollettino U. M. I. (7) 3-B (1989) 455–476.MathSciNetGoogle Scholar
  13. [AMZ 2]
    D.D. Anderson, J. Mott and M. Zafrullah, “Finite character representations for integral domains” Bollettino U.M.I (7) 6-B (1992) 613–630.MathSciNetGoogle Scholar
  14. [AMZ 3]
    D.D. Anderson, J. Mott and M. Zafrullah, “Unique factorization in non atomic domains” Bollettino U. M. I. (to appear)Google Scholar
  15. [AZ 1]
    D.D. Anderson and M. Zafrullah, “Weakly factorial domains and groups of divisibility” Proc. Amer. Math. Soc. 109(4)(1990) 907–913.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [AZ 2]
    D.D. Anderson and M. Zafrullah, “Independent locally-finite intersections of localizations” Houston J. Math. 25(3) (1999) 433–452.MathSciNetzbMATHGoogle Scholar
  17. [An 1]
    D.F. Anderson, “A general theory of class groups” Comm. Algebra 16(1988), 805–847MathSciNetzbMATHCrossRefGoogle Scholar
  18. [An 2]
    D. F. Anderson, “The class group and local class group of an integral domain” This ollectionGoogle Scholar
  19. [AEK]
    D.F. Anderson, S. Elbaghdadi and S. Kabbaj, “The class group of A + XB[X] domains” Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, New York, (1999) 73–85.Google Scholar
  20. [ANeA]
    D.F. Anderson and D. Nour El Abidine, “Factorization in integral domains, III” J. Pure Appl. Algebra 135(1999) 107–127.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [AR]
    D.F. Anderson and A. Rykaert, “The class group of D + M” J. Pure Appl. Algebra 52(1988) 199–212.MathSciNetzbMATHCrossRefGoogle Scholar
  22. [AB]
    J. Arnold and J. Brewer, “On flat overrings, ideal transforms and generalized trans-forms of a commutative ring” J. Algebra 18(1971) 254–263MathSciNetzbMATHCrossRefGoogle Scholar
  23. [Aub]
    K. Aubert, “Localisation dans les systemes d’ideaux” C. R. Acad. Sei. Paris 272(1971) 456–468.Google Scholar
  24. [BGR]
    V. Barucci, S. Gabelli and M. Roitman, “The class group of a strongly Mori domain” Comm. Algebra 22 (1994), no. 1, 173–211MathSciNetzbMATHCrossRefGoogle Scholar
  25. [BIK]
    V. Barucci, L. Izelgue, and S. Kabbaj, “Some factorization properties on A + XB[X] domains” Lecture Notes in Pure and Applied Mathematics, Marcel Dekker (1997) 69–78.Google Scholar
  26. [BG]
    E. Bastida and R. Gilmer, “Overrings and divisorial ideals of rings of the form D+M” Michigan Math. J. 20(1973) 79–95.MathSciNetzbMATHGoogle Scholar
  27. [Bou]
    A. Bouvier, “Le groupe des classes d’un anneau integre” 107 eme Congres National des Societe Savantes, Brest, Fasc. IV(1982) 85–92.Google Scholar
  28. [BZ 1]
    A. Bouvier and M. Zafrullah, “On the class group” (unpublished manuscript).Google Scholar
  29. [BZ 2]
    A. Bouvier and M. Zafrullah, “On some class groups of an integral domain” Bull. Soc. Math. Grece 29(1988) 45–59.MathSciNetzbMATHGoogle Scholar
  30. [BH]
    J. Brewer and W. Heinzer, “Associated primes of principal ideals” Duke Math. J. 41(1974) 1–7.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [BR]
    J. Brewer and E. Rutter, “D + M construction with general overrings” Michigan Math. J. 23(1976) 33–42.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [CGH]
    P.-J. Cahen, S. Gabelli and E.G. Houston, “Mori domains of integer-valued olynomials” preprint.Google Scholar
  33. [CMZ 1]
    D.L. Costa, J.L. Mott and M. Zafrullah, “The construction D + XD S[X]” J. Algebra 53(1978) 423–439.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [CMZ 2]
    D.L. Costa, J.L. Mott and M. Zafrullah, “Overrings and dimensions of general D + M constructions” J. Natur. Sei. and Math. 26(2) (1986), 7–14MathSciNetzbMATHGoogle Scholar
  35. [DHLZ]
    D. Dobbs, E. Houston, T. Lucas, E. Houston and M. Zafrullah, “t-linked overrings and Prüfer v-multiplication domains” Comm. Algebra 17(1989) 2835–2852.MathSciNetzbMATHCrossRefGoogle Scholar
  36. [DRSS]
    T. Dumitrescu, N. Radu, S. Al-Salihi and T. Shah, “Some factorization properties of composite domains A + XB[X] and A + XB[[X]]” Comm. Algebra (to appear).Google Scholar
  37. [FG]
    M. Fontana and S. Gabelli, “On the class group and the local class group of a pullback” J. Algebra 181(1996) 803–835.MathSciNetzbMATHCrossRefGoogle Scholar
  38. [FGH]
    M. Fontana, S. Gabelli and E. Houston, “UMT-domains and domains with Prüfer integral closure” Comm. Algebra 26(4)(1998) 1017–1039.MathSciNetzbMATHCrossRefGoogle Scholar
  39. [FHP]
    M. Fontana, J. Huckaba and I. Papick, Prüfer domains, Monographs and Textbooks in Pure and Applied Mathematics, 203, Marcel Dekker, Inc., New York, 1997.Google Scholar
  40. [Fos]
    R. Fossum, The divisor class group of a Krull domain, Ergebnisse der Mathematik und ihrer grenzgebiete B. 74, Springer-Verlag, Berlin, Heidelberg, New York, 1973.CrossRefGoogle Scholar
  41. [Gab 1]
    S. Gabelli, “On divisorial ideals in polynomial rings over Mori domains” Comm Algebra 15(11)(1987) 2349–2370.MathSciNetzbMATHCrossRefGoogle Scholar
  42. [Gab 2]
    S. Gabelli, “On Nagata’s theorem for the class group II” Lecture Notes in Pure and Applied Mathematics, Marcel-Dekker, New York (Commutative Algebra and Algebraic Geometry) 206(1999) 117–142.MathSciNetGoogle Scholar
  43. [GR]
    S. Gabelli and M. Roitman, “On Nagata’s theorem for the class group” J. Pure Appl. Algebra 66(1990) 31–42.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [Ger]
    A. Geroldinger, “Chains of factorizations and sets of lengths” J. Algebra 188(1997) 331–362MathSciNetzbMATHCrossRefGoogle Scholar
  45. [Gil]
    R. Gilmer, Multiplicative Ideal Theory, Marcel-Dekker, New York, 1972.zbMATHGoogle Scholar
  46. [GMZ]
    R. Gilmer, J. Mott and M. Zafrullah, “On t-invertibility and comparability” Lecture Notes in Pure and Applied Mathematics, Marcel-Dekker, New York (Proceedings of the Fes Conference) 153(1994) 141–150.MathSciNetGoogle Scholar
  47. [GV]
    S. Glaz and W. Vasconcelos, “Flat ideals II” Manuscripta Math. 22(1977) 325–341MathSciNetzbMATHCrossRefGoogle Scholar
  48. [Gri 1]
    M. Griffin, “Some results on v-multiplication rings” Canad. J. Math. 19(1967) 710–722.MathSciNetzbMATHCrossRefGoogle Scholar
  49. [Gri 2]
    M. Griffin, “Rings of Krull type” J. Reine Angew. Math. 229(1968) 1–27MathSciNetzbMATHGoogle Scholar
  50. [HH]
    J.R. Hedstrom and E.G. Houston, “Some remarks on star-operations” J. Pure Appl. Algebra 18(1980) 37–44.MathSciNetzbMATHCrossRefGoogle Scholar
  51. [H-K]
    F. Halter-Koch, Ideal Systems, An introduction to ideal theory, Marcel Dekker, New York, 1998.zbMATHGoogle Scholar
  52. [Hou]
    E. Houston, “On divisorial prime ideals in Prüfer v-multiplication domains” J. Pure Appl. Algebra 42(1986) 55–62.MathSciNetzbMATHCrossRefGoogle Scholar
  53. [HMM]
    E. Houston, S. Malik and J. Mott, “Characterizations of *-multiplication domains” Canad. Math. Bull. 27(1)(1984) 48–52.MathSciNetzbMATHCrossRefGoogle Scholar
  54. [HZ 1]
    E. Houston and M. Zafrullah, “Integral domains in which every t-ideal is divisorial” Michigan Math. J. 35(1988) 291–300.MathSciNetzbMATHCrossRefGoogle Scholar
  55. [HZ 2]
    E. Houston and M. Zafrullah, “On t-invertibility II” Comm. Algebra 17(8)(1989) 1955–1969.MathSciNetzbMATHCrossRefGoogle Scholar
  56. [Jaf]
    P. Jaffard, Les systemes d’ideaux, Dunod, Paris, 1960.zbMATHGoogle Scholar
  57. [Kai]
    F. Kainrath, “A divisor theoretic approach towards the arithmetic of Noetherian omains” (preprint)Google Scholar
  58. [Kan]
    B.G. Kang, “Prüfer v-multiplication domains and the ring R[X]n v” J. Algebra 123(1989) 151–170.MathSciNetzbMATHCrossRefGoogle Scholar
  59. [Kang]
    B.G. Kang, “On the converse of a well-known fact about Krull domains” J. Algebra 124(1989) 284–289.MathSciNetzbMATHCrossRefGoogle Scholar
  60. [Lor]
    P. Lorenzen, “Abstrakte Begrundung der multiplicativen Idealtheorie” Math. Z. 45(1939) 533–553.MathSciNetCrossRefGoogle Scholar
  61. [MMZ]
    S. Malik, J. Mott and M. Zafrullah, “On t-invertibility” Comm. Algebra 16(1988) 149–170.MathSciNetzbMATHCrossRefGoogle Scholar
  62. [Mat]
    E. Matlis, Torsion free Modules, The University of Chicago Press, Chicago-London, 1972.zbMATHGoogle Scholar
  63. [MZ]
    J. Mott and M. Zafrullah, “On Prüfer v-multiplication domains” Manuscripta Math. 35(1981)1–26.MathSciNetzbMATHCrossRefGoogle Scholar
  64. [Nag 1]
    M. Nagata, “On Krull’s conjecture concerning valuation rings” Nagoya Math. J. (4)(1952) 29–33.Google Scholar
  65. [Nag 2]
    M. Nagata, “Correction to my paper “On Krull’s conjecture concerning valuation overrings”” Nagoya Math.J. (9)(1955) 209–212.MathSciNetzbMATHGoogle Scholar
  66. [Nis 1]
    T. Nishimura, “On the v-ideal of an integral domain” Bull. Kyoto Gakugei Univ. (ser. B) 17(1961) 47–50.Google Scholar
  67. [Nis 2]
    T. Nishimura, “Unique factorization of ideals in the sense of quasi-equality”. J. Math. Kyoto Univ. (3)(1963) 115–125.MathSciNetzbMATHGoogle Scholar
  68. [Pic]
    G. Picavet, “About GCD domains” Lecture Notes in Pure and Applied Mathematics, Marcel Dekker volume 205(1999) 501–519.MathSciNetGoogle Scholar
  69. [P-H]
    M. Picavet-L’Hermitte, “Factorization in some orders with a PID as integral closure” Proceedings of the Number Theory Conference in Graz (held 1998), W. de Gruyter (to appear)Google Scholar
  70. [Que 1]
    J. Querre, “Sur une propriete des anneaux de Krull” Bull. Soc. Math. 2c serie 5(1971) 341–354.MathSciNetGoogle Scholar
  71. [Que 2]
    J. Querre, “Sur les anneaux reflexifs” Can. J. Math. 6(1975) 1222–1228.MathSciNetCrossRefGoogle Scholar
  72. [Rai]
    N. Raillard, “Sur les anneaux de Mori” C. R. Acad. Sci. Paris Sér. A-B 280 (1975), no.23, Ai, A1571-A1573.MathSciNetGoogle Scholar
  73. [Rib]
    P. Ribenboim, “Anneaux normaux reels a caractere fini” Summa Brasil. Math. 3(1956) 213–253.MathSciNetGoogle Scholar
  74. [Ryk]
    A. Rykaert, “Sur le groupe des classes et le groupe local des classes d’une anneau integre”, Thesis, Universite Claude Bernard de Lyon I, 1986.Google Scholar
  75. [Tan]
    H. Tang, “Gauss Lemma” Proc. Amer. Math. Soc. 35(1972) 372–376.MathSciNetzbMATHGoogle Scholar
  76. [Wan 1]
    F. Wang, “On w-projective modules and w-flat modules” Algebra Colloq. 4(1)(1997) 111–120.MathSciNetzbMATHGoogle Scholar
  77. [Wan 2]
    F. Wang, “A note on two-dimensional rings” Comm. Algebra 27(1) (1999) 405–409.MathSciNetzbMATHCrossRefGoogle Scholar
  78. [Wan 3]
    F. Wang, “w;-dimension of domains” Comm. Algebra 27(5) (1999) 2267–2276.MathSciNetzbMATHCrossRefGoogle Scholar
  79. [Wan 4]
    F. Wang, “w-modules over PVMD’s” (preprint).Google Scholar
  80. [Wan 5]
    F. Wang, “On UMT domains and w-integral dependence” (preprint).Google Scholar
  81. [WM]
    F. Wang and R. McCasland, “On w-modules over strong Mori domains” Comm. Algebra 25(1997) 1285–1306.MathSciNetzbMATHCrossRefGoogle Scholar
  82. [Zafr]
    M. Zafrullah, “Rigid elements in GCD domains” J. Natur. Sci. and Math. 17 (1977) 7–14.MathSciNetzbMATHGoogle Scholar
  83. [Zaf 1]
    M. Zafrullah, “On a result of Gilmer” J. London Math. Soc. 16(1977) 19–20.MathSciNetCrossRefGoogle Scholar
  84. [Zaf 2]
    M. Zafrullah, “Finite conductor domains” Manuscripta Math. 24(1978) 191–203.MathSciNetzbMATHCrossRefGoogle Scholar
  85. [Zaf 3]
    M. Zafrullah, “A general theory of almost factoriality” Manuscripta Math. 15(1985) 29–62.MathSciNetCrossRefGoogle Scholar
  86. [Zaf 4]
    M. Zafrullah, “The D + XD S[X] construction from GCD-domains” J. Pure Appl. Algebra 50(1988) 93–107MathSciNetzbMATHCrossRefGoogle Scholar
  87. [Zaf 5]
    M. Zafrullah, “Some polynomial characterizations of Prüfer v-multiplication domains” J. Pure Appl. Algebra 32(1984) 231–237.MathSciNetzbMATHCrossRefGoogle Scholar
  88. [Zaf 6]
    M. Zafrullah, “The v-operation and intersections of quotient rings of integral domains” Comm. Algebra 13(1985) 1699–1712.MathSciNetzbMATHCrossRefGoogle Scholar
  89. [Zaf 7]
    M. Zafrullah, “Ascending chain conditions and star operations”, Comm. Algebra 17(6)(1989) 1523–1533.MathSciNetzbMATHCrossRefGoogle Scholar
  90. [Zaf 8]
    M. Zafrullah, “Well behaved prime t-ideals” J. Pure Appl. Algebra 65(1990) 199–207.MathSciNetzbMATHCrossRefGoogle Scholar
  91. [Zaf 9]
    M. Zafrullah, “Flatness and invertibility of an ideal” Comm. Algebra 18(7)(1990) 2151–2158.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Muhammad Zafrullah
    • 1
  1. 1.Department of Mathematics, SCEN 301The University of ArkansasFayettevilleUSA

Personalised recommendations