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Open Problems in Commutative Ring Theory

  • Paul-Jean Cahen
  • Marco Fontana
  • Sophie Frisch
  • Sarah Glaz
Chapter

Abstract

This chapter consists of a collection of open problems in commutative algebra. The collection covers a wide range of topics from both Noetherian and non-Noetherian ring theory and exhibits a variety of research approaches, including the use of homological algebra, ring theoretic methods, and star and semistar operation techniques. The problems were contributed by the authors and editors of this volume, as well as other researchers in the area.

Keywords

Prüfer ring Homological dimensions Integral closure Group ring Grade Complete ring McCoy ring Straight domain Divided domain Integer-valued polynomials Factorial Density Matrix ring Overring Absorbing ideal Kronecker function ring Stable ring Divisorial domain Mori domain Finite character PvMD Semistar operation Star operation Jaffard domain Locally tame domain Factorization Spectrum of a ring Integral closure of an ideal Rees algebra Rees valuation 

Mathematics Subject Classification (2010):

13-02 13A05 13A15 13A18 13B22 13C15 13D05 13D99 13E05 13F05 13F20 13F30 13G05 

Notes

Acknowledgements

We thank all the commutative algebraists who contributed open problems to this chapter. The list of contributors is as follows: D.D. Anderson (Problem 8), A. Badawi (Problem 30), P.-J. Cahen (Problems 14 and 15), J.-L. Chabert (Problems 16–18), J. Elliott (Problems 19–23), C.A. Finocchiaro and M. Fontana (Problem 36), S. Frisch (Problems 28 and 39), S. Gabelli (Problems 32 and 33), A. Geroldinger (Problem 38), S. Glaz (Problems 1–3), L. Hummel (Problem 7), K. Johnson (Problems 24 and 25), S. Kabbaj (Problem 37), T.G. Lucas (Problems 9–12), B. Olberding (Problems 29 and 31), G. Peruginelli (Problem 26), G. Picavet and M. Picavet-L’Hermitte (Problem 13), R. Schwarz (Problems 4–6), I. Swanson (Problems 41–44), N.J. Werner (Problems 27 and 28), S. Wiegand and R. Wiegand (Problem 40), M. Zafrullah (Problems 34 and 35).

Note that while this chapter and reference [54] were in proof, Problem 32a has been answered negatively. A counter example is given in [54, Example 3.9] with a 2-dimensional Prüfer domain.

References

  1. 1.
    D. Adam, J.-L. Chabert, Y. Fares, Subsets of Z with simultaneous ordering. Integers 10, 437–451 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    T. Akiba, Integrally-closedness of polynomial rings. Jpn. J. Math. 6, 67–75 (1980)zbMATHMathSciNetGoogle Scholar
  3. 3.
    D.D. Anderson, Star operations induced by overrings. Comm. Algebra 16, 2535–2553 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    D.D. Anderson, Quasi-complete semilocal rings and modules, in Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, ed. by M. Fontana, S. Frisch, S. Glaz (Springer, New York, 2014, this volume)Google Scholar
  5. 5.
    D.D. Anderson, M. Zafrullah, Almost Bézout domains. J. Algebra 142, 285–309 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    D.D. Anderson, M. Zafrullah, Integral domains in which nonzero locally principal ideals are invertible. Comm. Algebra 39, 933–941 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    D.F. Anderson, A. Badawi, On n-absorbing ideals of commutative rings. Comm. Algebra 39, 1646–1672 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    D.F. Anderson, D.E. Dobbs, P.M. Eakin, W.J. Heinzer, On the generalized principal ideal theorem and Krull domains. Pacific J. Math. 146(2), 201–215 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    D.D. Anderson, D.F. Anderson, M. Zafrullah, The ring D + XD S[X] and t-splitting sets. Comm. Algebra Arab. J. Sci. Eng. Sect. C Theme Issues 26(1), 3–16 (2001)zbMATHMathSciNetGoogle Scholar
  10. 10.
    M. André, Non-Noetherian complete intersections. Bull. Am. Math. Soc. 78, 724–729 (1972)CrossRefzbMATHGoogle Scholar
  11. 11.
    A. Badawi, On 2-absorbing ideals of commutative rings. Bull. Austral. Math. Soc. 75, 417–429 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    S. Bazzoni, Class semigroups of Prüfer domains. J. Algebra 184, 613–631 (1996)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    S. Bazzoni, Finite character of finitely stable domains. J. Pure Appl. Algebra 215, 1127–1132 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    S. Bazzoni, S. Glaz, Gaussian properties of total rings of quotients. J. Algebra 310, 180–193 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    M. Bhargava, The factorial function and generalizations. Am. Math. Monthly 107, 783–799 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    S. Bouchiba, S. Kabbaj, Bouvier’s Conjecture. Commutative Algebra and its Applications (de Gruyter, Berlin, 2009), pp. 79–88Google Scholar
  17. 17.
    A. Bouvier, The local class group of a Krull domain. Canad. Math. Bull. 26, 13–19 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    A. Bouvier, S. Kabbaj, Examples of Jaffard domains. J. Pure Appl. Algebra 54(2–3), 155–165 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    P.-J. Cahen, J.-L. Chabert, Integer-Valued Polynomials. Mathematical Surveys and Monographs, vol. 48 (American Mathematical Society, Providence, 1997)Google Scholar
  20. 20.
    P.-J. Cahen, R. Rissner, Finiteness and Skolem closure of ideals for non unibranched domains. Comm. Algebra, in PressGoogle Scholar
  21. 21.
    E. Celikbas, C. Eubanks-Turner, S. Wiegand, Prime ideals in polynomial and power series rings over Noetherian domains, in Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, ed. by M. Fontana, S. Frisch, S. Glaz (Springer, New York, 2014, this volume)Google Scholar
  22. 22.
    J.-L. Chabert, Integer-valued polynomials: looking for regular bases (a survey), in Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, ed. by M. Fontana, S. Frisch, S. Glaz (Springer, New York, 2014, this volume)Google Scholar
  23. 23.
    S. Chapman, S. Glaz, One hundred problems in commutative ring theory, in Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications, vol. 520 (Kluwer Academic Publishers, Dordrecht, 2000), pp. 459–476Google Scholar
  24. 24.
    C. Chevalley, On the theory of local rings. Ann. Math. 44, 690–708 (1943)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Z. Coelho, W. Parry, Ergodicity of p-adic multiplications and the distribution of Fibonacci numbers. Am. Math. Soc. Translations 202, 51–70 (2001)MathSciNetGoogle Scholar
  26. 26.
    D. Costa, J.L. Mott, M. Zafrullah, The construction D + XD S[X]. J. Algebra 53, 423–439 (1978)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    H. Coughlin, Classes of normal monomial ideals, Ph.D. thesis, University of Oregon, 2004Google Scholar
  28. 28.
    S.D. Cutkosky, On unique and almost unique factorization of complete ideals II. Invent. Math. 98, 59–74 (1989)CrossRefzbMATHMathSciNetGoogle Scholar
  29. 29.
    J. David, A characteristic zero non-Noetherian factorial ring of dimension three. Trans. Am. Math. Soc. 180, 315–325 (1973)CrossRefzbMATHGoogle Scholar
  30. 30.
    D. Dobbs, G. Picavet, Straight rings. Comm. Algebra 37(3), 757–793 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    D.E. Dobbs, M. Fontana, S. Kabbaj, Direct limits of Jaffard domains and S-domains. Comment. Math. Univ. St. Pauli 39(2), 143–155 (1990)zbMATHMathSciNetGoogle Scholar
  32. 32.
    D. Dobbs, G. Picavet, M. Picavet-L’Hermitte, On a new class of integral domains with the portable property, in Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, ed. by M. Fontana, S. Frisch, S. Glaz (Springer, New York, 2014, this volume)Google Scholar
  33. 33.
    A. Douglas, The weak global dimension of the group rings of abelian groups. J. London Math. Soc. 36, 371–381 (1961)CrossRefzbMATHMathSciNetGoogle Scholar
  34. 34.
    T. Dumitrescu, Y. Lequain, J. Mott, M. Zafrullah, Almost GCD domains of finite t-character. J. Algebra 245, 161–181 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  35. 35.
    J. Elliott, Universal properties of integer-valued polynomial rings. J. Algebra 318, 68–92 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  36. 36.
    J. Elliott, Birings and plethories of integer-valued polynomials, in Third International Meeting on Integer-Valued Polynomials, 2010. Actes des Rencontres du CIRM, vol. 2(2) (CIRM, Marseille, 2010), pp. 53–58Google Scholar
  37. 37.
    J. Elliott, Integer-valued polynomial rings, t-closure, and associated primes. Comm. Algebra 39(11), 4128–4147 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  38. 38.
    J. Elliott, The probability that Intn(D) is free, in Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, ed. by M. Fontana, S. Frisch, S. Glaz (Springer, New York, 2014, this volume)Google Scholar
  39. 39.
    A. Fabbri, O. Heubo-Kwegna, Projective star operations on polynomial rings over a field. J. Comm. Algebra 4(3), 387–412 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  40. 40.
    C.A. Finocchiaro, G. Picozza, F. Tartarone, Star-invertibility and t-finite character in integral domains. J. Algebra Appl. 10, 755–769 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  41. 41.
    C.A. Finocchiaro, M. Fontana, K.A. Loper, The constructible topology on spaces of valuation domains. Trans. Am. Math. Soc. 365(12), 6199–6216 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  42. 42.
    M. Fontana, J. Huckaba, Localizing systems and semistar operations, in Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications, vol. 520 (Kluwer Academic Publishers, Dordrecht, 2000), pp. 169–197Google Scholar
  43. 43.
    M. Fontana, S. Kabbaj, Essential domains and two conjectures in dimension theory. Proc. Am. Math. Soc. 132, 2529–2535 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  44. 44.
    M. Fontana, M. Zafrullah, On v-domains: a survey, in Commutative Algebra: Noetherian and Non-Noetherian Perspectives (Springer, New York, 2011), pp. 145–179Google Scholar
  45. 45.
    M. Fontana, E. Houston, T.G. Lucas, Toward a classification of prime Ideals in Prüfer domains. Forum Mathematicum, 22, 741–766 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  46. 46.
    S. Frisch, Polynomial separation of points in algebras, in Arithmetical Properties of Commutative Rings and Modules (Chapel Hill Conference Proceedings) (Dekker, New York, 2005), pp. 249–254Google Scholar
  47. 47.
    S. Frisch, A Construction of integer-valued polynomials with prescribed sets of lengths of factorizations. Monatsh. Math. 171(3–4), 341–350 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  48. 48.
    S. Frisch, Corrigendum to Integer-valued polynomials on algebras. [J. Algebra 373, 414–425] (2013), J. Algebra 412 282 (2014)Google Scholar
  49. 49.
    S. Gabelli, Locally principal ideals and finite character. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 56(104), 99–108 (2013)Google Scholar
  50. 50.
    S. Gabelli, Ten problems on stability of domains, in Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, ed. by M. Fontana, S. Frisch, S. Glaz (Springer, New York, 2014, this volume)Google Scholar
  51. 51.
    S. Gabelli, E. Houston, Coherent-like conditions in pullbacks. Michigan Math. J. 44, 99–123 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  52. 52.
    S. Gabelli, G. Picozza, Star-stable domains. J. Pure Appl. Algebra 208, 853–866 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  53. 53.
    S. Gabelli, G. Picozza, Star stability and star regularity for Mori domains. Rend. Semin. Mat. Padova, 126, 107–125 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  54. 54.
    S. Gabelli, M. Roitman, On finitely stable domains (preprint)Google Scholar
  55. 55.
    W. Gao, A. Geroldinger, W.A. Schmid, Local and global tameness in Krull monoids. Comm. Algebra (to appear) http://arxiv.org/abs/1302.3078
  56. 56.
    A. Geroldinger, F. Halter-Koch, Non-Unique Factorizations: Algebraic, Combinatorial and Analytic Theory. Pure and Applied Mathematics, vol. 278 (Chapman & Hall/CRC Press, Boca Raton, 2006)Google Scholar
  57. 57.
    A. Geroldinger, W. Hassler, Local tameness of v-Noetherian monoids. J. Pure Appl. Algebra 212, 1509–1524 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  58. 58.
    A. Geroldinger, W. Hassler, G. Lettl, On the arithmetic of strongly primary monoids. Semigroup Forum 75, 567–587 (2007)CrossRefMathSciNetGoogle Scholar
  59. 59.
    R. Gilmer, Multiplicative Ideal Theory (Dekker, New York, 1972)zbMATHGoogle Scholar
  60. 60.
    R. Gilmer, Commutative Semigroup Rings. Chicago Lecture Notes in Mathematics (University of Chicago Press, Chicago, 1984)Google Scholar
  61. 61.
    R. Gilmer, T. Parker, Divisibility properties in semigroup rings. Mich. Math. J. 21, 65–86 (1974)CrossRefzbMATHMathSciNetGoogle Scholar
  62. 62.
    S. Glaz, On the weak dimension of coherent group rings. Comm. Algebra. 15, 1841–1858 (1987)CrossRefzbMATHMathSciNetGoogle Scholar
  63. 63.
    S. Glaz, Commutative Coherent Rings. Lecture Notes in Mathematics, vol. 1371 (Springer, Berlin, 1989)Google Scholar
  64. 64.
    S. Glaz, Finite conductor rings. Proc. Am. Math. Soc. 129, 2833–2843 (2000)CrossRefMathSciNetGoogle Scholar
  65. 65.
    S. Glaz, Finite conductor rings with zero divisors, in Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications, vol. 520 (Kluwer Academic Publishers, Dordrecht, 2000), pp. 251–270Google Scholar
  66. 66.
    S. Glaz, Prüfer Conditions in Rings with Zero-Divisors. Lecture Notes in Pure and Applied Mathematics, vol. 241 (CRC Press, London, 2005), pp. 272–281Google Scholar
  67. 67.
    S. Glaz, The weak dimension of Gaussian rings. Proc. Am. Math. Soc. 133, 2507–2513 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  68. 68.
    S. Glaz, R. Schwarz, Prüfer conditions in commutative rings. Arabian J. Sci. Eng. 36, 967–983 (2011)CrossRefMathSciNetGoogle Scholar
  69. 69.
    S. Glaz, R. Schwarz, Finiteness and homological conditions in commutative group rings, in Progress in Commutative Algebra 2 (De Gruyter, Berlin, 2012), pp. 129–143Google Scholar
  70. 70.
    F. Halter-Koch, Kronecker function rings and generalized integral closures. Comm. Algebra 31, 45–59 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  71. 71.
    F. Halter-Koch, Clifford semigroups of ideals in monoids and domains. Forum Math. 21, 1001–1020 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  72. 72.
    T. Hamilton, T. Marley, Non-Noetherian Cohen-Macaulay rings. J. Algebra 307, 343–360 (2007)zbMATHMathSciNetGoogle Scholar
  73. 73.
    W. Hassler, Arithmetical properties of one-dimensional, analytically ramified local domains. J. Algebra 250, 517–532 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  74. 74.
    W. Heinzer, C. Rotthaus, S. Wiegand, Examples using power series over Noetherian integral domains. http://www.math.purdue.edu/~heinzer/preprints/preprints.html (in preparation)
  75. 75.
    M. Hochster, Presentation depth and the Lipman-Sathaye Jacobian theorem, The Roos Festschrift, vol. 2, Homology Homotopy Appl. 4, 295–314 (2002)Google Scholar
  76. 76.
    W.C. Holland, J. Martinez, W.Wm. McGovern, M. Tesemma, Bazzoni’s conjecture. J. Algebra 320(4), 1764–1768 (2008)Google Scholar
  77. 77.
    E. Houston, M. Zafrullah, Integral domains in which each t-ideal is divisorial. Michigan Math J. 35(2), 291–300 (1988)CrossRefzbMATHMathSciNetGoogle Scholar
  78. 78.
    J.A. Huckaba, Commutative Rings with Zero Divisors (Dekker, New York, 1988)zbMATHGoogle Scholar
  79. 79.
    L. Hummel, Recent progress in coherent rings: a homological perspective, in Progress in Commutative Algebra 1 (De Gruyter, Berlin, 2012), pp. 271–292Google Scholar
  80. 80.
    L. Hummel, T. Marley, The Auslander-Bridger formula and the Gorenstein property for coherent rings. J. Comm. Algebra. 1, 283–314 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  81. 81.
    C. Huneke, I. Swanson, Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336 (Cambridge University Press, Cambridge, 2006)Google Scholar
  82. 82.
    C.J. Hwang, G.W. Chang, Prüfer v-multiplication domains in which each t-ideal is divisorial. Bull. Korean Math. Soc. 35(2), 259–268 (1998)zbMATHMathSciNetGoogle Scholar
  83. 83.
    K. Johnson, K. Scheibelhut, Polynomials that are integer valued on the Fibonacci numbers (to appear)Google Scholar
  84. 84.
    G. Karpilovsky, Commutative Group Algebras. Lecture Notes in Pure and Applied Mathematics, vol. 78. (Dekker, New York, 1983)Google Scholar
  85. 85.
    T.Y. Lam, A First Course in Non-Commutative Rings (Springer, New York, 1991)CrossRefGoogle Scholar
  86. 86.
    D.A. Leonard, R. Pellikaan, Integral closures and weight functions over finite fields. Finite Fields Appl. 9, 479–504 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  87. 87.
    A.K. Loper, A classification of all D such that Int(D) is a Prüfer domain. Proc. Am. Math Soc. 126(3), 657–660 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  88. 88.
    K.A. Loper, F. Tartarone, A classification of the integrally closed rings of polynomials containing Z[X]. J. Comm. Algebra 1(1), 91–157 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  89. 89.
    K.A. Loper, N.J. Werner, Generalized rings of integer-valued polynomials. J. Num. Theory 132(11), 2481–2490 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  90. 90.
    T.G. Lucas, Two annihilator conditions: property (A) and (a.c.). Comm. Algebra 14, 557–580 (1986)Google Scholar
  91. 91.
    W.Wm. McGovern, Prüfer domains with Clifford class semigroup. J. Comm. Algebra 3, 551–559 (2011)Google Scholar
  92. 92.
    A. Mingarelli, Abstract factorials. arXiv:0705.4299v3[math.NT]. Accessed 10 July 2012Google Scholar
  93. 93.
    J.L. Mott, M. Zafrullah, On Krull domains. Arch. Math. 56, 559–568 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  94. 94.
    M. Nagata, On the fourteenth problem of Hilbert, in Proceedings of the International Congress of Mathematicians, 1958 (Cambridge University Press, London-New York, 1960), pp. 459–462Google Scholar
  95. 95.
    A. Okabe, R. Matsuda, Semistar operations on integral domains. Math. J. Toyama Univ. 17, 1–21 (1994)zbMATHMathSciNetGoogle Scholar
  96. 96.
    B. Olberding, Intersections of Valuation Overrings of Two-Dimensional Noetherian Domains. Commutative Algebra–Noetherian and Non-Noetherian Perspectives (Springer, New York, 2011), pp. 459–462Google Scholar
  97. 97.
    B. Osofsky, Global dimensions of commutative rings with linearly ordered ideals. J. Lond. Math. Soc. 44, 183–185 (1969)CrossRefzbMATHMathSciNetGoogle Scholar
  98. 98.
    G. Peruginelli, Integer-valued polynomials over matrices and divided differences. Monatshefte für Mathematik. http://dx.doi.org/10.1007/s00605-013-0519-9 (to appear)
  99. 99.
    G. Peruginelli, N.J. Werner, Integral closure of rings of integer-valued polynomials on algebras, in Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, ed. by M. Fontana, S. Frisch, S. Glaz (Springer, New York, 2014, this volume)Google Scholar
  100. 100.
    D. Rees, On a problem of Zariski. Illinois J. Math. 2, 145–149 (1958)zbMATHMathSciNetGoogle Scholar
  101. 101.
    P. Roberts, An infinitely generated symbolic blow-up in a power series ring and a new counterexample to Hilbert’s fourteenth problem. J. Algebra 132, 461–473 (1990)CrossRefzbMATHMathSciNetGoogle Scholar
  102. 102.
    K. Scheibelhut, Polynomials that are integer valued on the Fibonacci numbers, M.Sc. Thesis, Dalhousie University, 2013Google Scholar
  103. 103.
    R. Schwarz, S. Glaz, Commutative group rings with von Neumann regular total rings of quotients. J. Algebra 388, 287–293 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  104. 104.
    A.K. Singh, I. Swanson, Associated primes of local cohomology modules and of Frobenius powers. Int. Math. Res. Notices 30, 1703–1733 (2004)CrossRefMathSciNetGoogle Scholar
  105. 105.
    I. Swanson, Integral closure, in Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, ed. by M. Fontana, S. Frisch, S. Glaz (Springer, New York, 2014, this volume)Google Scholar
  106. 106.
    N.J. Werner, Integer-valued polynomials over matrix rings. Comm. Algebra 40(12), 4717–4726 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  107. 107.
    N.J. Werner, Polynomials that kill each element of a finite ring. J. Algebra Appl. 13, 1350111 (2014) http://dx.doi.org/10.1142/S0219498813501119
  108. 108.
    R. Wiegand, The prime spectrum of a two-dimensional affine domain. J. Pure Appl. Algebra 40, 209–214 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  109. 109.
    R. Wiegand, S. Wiegand, Prime ideals and decompositions of modules, in Non-Noetherian Commutative Ring Theory. Mathematics and Its Applications, vol. 520 (Kluwer Academic Publishers, Dordrecht, 2000), pp. 403–428Google Scholar
  110. 110.
    R. Wiegand, S. Wiegand, Prime Ideals in Noetherian Rings: A Survey. Ring and Module Theory (Birkhauser, Boston, 2010), pp. 175–193Google Scholar
  111. 111.
    M. Zafrullah, t-invertibility and Bazzoni-like statements. J. Pure Appl. Algebra 214, 654–657 (2010)Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Paul-Jean Cahen
    • 1
  • Marco Fontana
    • 2
  • Sophie Frisch
    • 3
  • Sarah Glaz
    • 4
  1. 1.Aix en ProvenceFrance
  2. 2.Dipartimento di Matematica e FisicaUniversità degli Studi Roma TreRomaItaly
  3. 3.Mathematics DepartmentGraz University of TechnologyGrazAustria
  4. 4.Department of MathematicsUniversity of ConnecticutStorrsUSA

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