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Commutative Rings of Dimension 0

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

Abstract

All rings considered in this paper are assumed to be commutative and unitary. If R is a subring of S, we assume that the unity element of S belongs to R, and hence is the unity of R. All allusions to the dimension of a ring refer to its Krull dimension. Thus dim R = n if there exists a chain P 0 < Pi< … < P n of proper prime ideals of R, but no longer such chain; dim R = ∞ if there exist arbitrarily long chains of prime ideals of R. This paper is concerned primarily with rings of dimension 0, hence rings whose proper prime ideals are maximal. There has been a surge of recent work in this area; the work seems to have originated with the papers [2]–[4] of M. Arapovic. The main focus of Arapovic’s work is the question of embeddability of a ring in a zero-dimensional ring, the subject of our Section 5. This paper contains an account of some of the work that has been done in the area of zero-dimensional commutative rings since the publication of Arapovic’s papers.

Keywords

Primary Ideal Commutative Ring Noetherian Ring Regular Ring Residue Field 
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.

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References

  1. [1]
    K. Aoyama, On the structure space of a direct product of rings, J. Sci. Hiroshima Univ. Ser. A-I, 34 (1970), 339–353.zbMATHGoogle Scholar
  2. [2]
    M. Arapovic, Characterizations of the 0-dimensional rings, Glas. Mat. 18 (1983), 39–46.MathSciNetGoogle Scholar
  3. [3]
    M. Arapovic, The minimal 0-dimensional overrings of commutative rings, Glas. Mat. 18 (1983), 47–52.MathSciNetGoogle Scholar
  4. [4]
    M. Arapovic, On the imbedding of a commutative ring into a *-dimensional ring, Glas. Mat. 18 (1983), 53–59.MathSciNetGoogle Scholar
  5. [5]
    G. Birkhoff, Lattice Theory, American Math. Soc. Colloq. Publ. Vol. 25, New York, 1948.zbMATHGoogle Scholar
  6. [6]
    G. Birkhoff and S. MacLane, A Survey of Modern Algebra, Macmillan, New York, 1941.zbMATHGoogle Scholar
  7. [7]
    G. Boole, The Mathematical Analysis of Logic, Cambridge, 1847.Google Scholar
  8. [8]
    A. M. S. Doering and Y. Lequain, The gluing of maximal idealsspectrum of a Noetherian ringgoing up and going down in polynomial rings, Trans. Amer. Math. Soc. 260 (1980), 583–593.MathSciNetCrossRefGoogle Scholar
  9. [9]
    R. Gilmer, Multiplicative Ideal Theory, Queen’s Papers Pure Appl. Math. Vol. 90, Kingston, Ontario, 1992.zbMATHGoogle Scholar
  10. [10]
    R. Gilmer, Background and preliminaries on zero-dimensional rings, Zero-dimensional Comutative Rings, Marcel Dekker, New York, 1995, pp. 1–13.Google Scholar
  11. [11]
    R. Gilmer, Zero-dimensionality and products of commutative rings, Zero-dimensional Comutative Rings, Marcel Dekker, New York, 1995, pp. 15–25.Google Scholar
  12. [12]
    R. Gilmer, Zero-dimensional extension rings and subrings, Zero-dimensional Comutative Rings, Marcel Dekker, New York, 1995, pp. 27–39.Google Scholar
  13. [13]
    R. Gilmer, Residue fields of zero-dimensional rings, Zero-dimensional Comutative Rings, Marcel Dekker, New York, 1995, pp. 41–55.Google Scholar
  14. [14]
    R. Gilmer and W. Heinzer, On the imbedding of a direct product into a zero-dimensional commutative ring, Proc. Amer. Math. Soc. 106 (1989), 631–637.MathSciNetzbMATHCrossRefGoogle Scholar
  15. [15]
    R. Gilmer and W. Heinzer, Products of commutative rings and zero-dimensionality, Trans. Amer. Math. Soc. 331 (1992), 663–680.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    R. Gilmer and W. Heinzer, Zero-dimensionality in commutative rings, Proc. Amer. Math. Soc. 115 (1992), 881–893.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    R. Gilmer and W. Heinzer, The family of residue fields of a zero-dimensional commutative ring, J. Pure Appl. Algebra 82 (1992), 131–153.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    R. Gilmer and W. Heinzer, Artinian subrings of a commutative ring, Trans. Amer. Math. Soc. 336 (1993), 295–310.MathSciNetzbMATHCrossRefGoogle Scholar
  19. [19]
    R. Gilmer and W. Heinzer, Imbeddability of a commutative ring in a finite-dimensional ring, manuscr. math. 84 (1994), 401–414.MathSciNetzbMATHCrossRefGoogle Scholar
  20. [20]
    R. Gilmer and W. Heinzer, Homomorphic images of an infinite product of zero-dimensional rings, Commun. Algebra 23 (1995), 1953–1965.MathSciNetzbMATHCrossRefGoogle Scholar
  21. [21]
    R. Gilmer and W. Heinzer, Infinite products of zero-dimensional commutative rings, Houston J. Math. 21 (1995), 247–259.MathSciNetzbMATHGoogle Scholar
  22. [22]
    R. Gilmer and W. Heinzer, On the embedding of a commutative ring in a local ring, Ill. J. Math. 43 (1999), 192–210.MathSciNetzbMATHGoogle Scholar
  23. [23]
    W. Heinzer, D. Lantz and R. Wiegand, The residue fields of a zero-dimensional ring, J. Pure Appl. Math. 129 (1998), 67–85.MathSciNetzbMATHGoogle Scholar
  24. [24]
    R. Heitmann, PID’s with specified residue class fields,Duke Math. J. 41 (1974), 565–582.MathSciNetzbMATHCrossRefGoogle Scholar
  25. [25]
    J. Huckaba, Commutative Rings with Zero Divisors, Marcel Dekker, New York, 1988.zbMATHGoogle Scholar
  26. [26]
    E. V. Huntington, Sets of independent postulates for the algebra of logic, Trans. Amer. Math. Soc. 5 (1904), 288–309.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [27]
    E. V. Huntington, New sets of independent postulates for the algebra of logic, with special reference to Whitehead and Russell’s Principia Mathematica, Trans. Amer. Math. Soc. 35 (1933), 274–304; 557–558; 971.MathSciNetGoogle Scholar
  28. [28]
    I. Kaplansky, Lattices of continuous functions, Bull. Amer. Math. Soc. 53 (1947), 617–622.MathSciNetzbMATHCrossRefGoogle Scholar
  29. [29]
    J. L. Kelley, General Topology, Van Nostrand, Princeton, NJ, 1955.zbMATHGoogle Scholar
  30. [30]
    J. W. Kerr, Very long chains of annihilator ideals, Israel J. Math. 46 (1983), 197–204.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [31]
    R. Levy, P. Loustanau, and J. Shapiro, The prime spectrum of an infinite product of copies of Z, Fund. Math. 138 (1991), 155–164.MathSciNetzbMATHGoogle Scholar
  32. [32]
    A. Magid, Direct limits of finite products of fields, Zero-dimensional Commutative Rings, Marcel Dekker, New York, 1995, pp. 299–305.Google Scholar
  33. [33]
    P. Maroscia, Sur les anneaux de dimension zero, Atti Accad. Naz. Lincei Rend. Cl. Sei. Fis. Mat. Natur. 56 (1974), 451–459.MathSciNetzbMATHGoogle Scholar
  34. [34]
    J. von Neumann, On regular rings, Proc. Nat. Acad. Sci. USA 22 (1936), 707–713.CrossRefGoogle Scholar
  35. [35]
    J. P. Olivier, Anneaux absolument plats universels et epimorphismes a buts reduits, Sem. Samuel, Paris, 1967–68.Google Scholar
  36. [36]
    N. Popescu and C. Vraciu, Sur la structure des anneaux absoluments plat commutatifs, J. Algebra 40 (1976), 364–383.MathSciNetzbMATHCrossRefGoogle Scholar
  37. [37]
    N. Popescu and C. Vraciu, Some remarks about the regular ring associated to a commutative ring, Rev. Roumaine Math. Pures Appl. 23 (1978), 269–277.MathSciNetzbMATHGoogle Scholar
  38. [38]
    J. Shapiro, The prime spectrum of an infinite product of zero-dimensional rings, Zero-dimensional Commutative Rings, Marcel Dekker, New York, 1995, pp. 347–356.Google Scholar
  39. [39]
    M. H. Stone, Boolean algebras and their relation to topology, Proc. Nat. Acad. Sci. USA 20 (1934), 197–202.CrossRefGoogle Scholar
  40. [40]
    M. H. Stone, The theory of representations for Boolean algebras, Trans. Amer. Math. Soc. 40 (1936), 37–111.MathSciNetGoogle Scholar
  41. [41]
    M. H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41 (1937), 375–481.MathSciNetCrossRefGoogle Scholar
  42. [42]
    M. H. Stone, The representation of Boolean algebras, Bull. Amer. Math. Soc. 44 (1938), 807–816.MathSciNetCrossRefGoogle Scholar
  43. [43]
    H. H. Storrer, Epimorphismen von kommutativen Ringen, Comment. Math. Helvetici 43 (1968), 378–401.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [44]
    R. Wiegand, Modules over universal regular rings, Pacific J. Math. 39 (1971), 807–819.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Robert Gilmer
    • 1
  1. 1.Department of MathematicsFlorida State UniversityTallahasseeUSA

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