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Topologically defined classes of going-down domains


Author: Ira J. Papick
Journal: Trans. Amer. Math. Soc. 219 (1976), 1-37
MSC: Primary 13G05
DOI: https://doi.org/10.1090/S0002-9947-1976-0401745-0
MathSciNet review: 0401745
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Abstract: Let R be an integral domain. Our purpose is to study GD (going-down) domains which arise topologically; that is, we investigate how certain going-down assumptions on R and its overrings relate to the topological space $ {\text{Spec}}(R)$. Many classes of GD domains are introduced topologically, and a systematic study of their behavior under homomorphic images, localization and globalization, integral change of rings, and the ``$ D + M$ construction'' is undertaken. Also studied, is the algebraic and topological relationships between these newly defined classes of GD domains.


References [Enhancements On Off] (What's this?)

  • [1] E. Bastida and R. Gilmer, Overrings and divisorial ideals of rings of the form $ D + M$, Michigan Math. J. 20 (1973), 79-95. MR 48 #2138. MR 0323782 (48:2138)
  • [2] N. Bourbaki, Commutative algebra, Addison-Wesley, Reading, Mass., 1972.
  • [3] J. Dawson and D. E. Dobbs, On going down in polynomial rings, Canad. J. Math. 26 (1974), 177-184. MR 48 #8490. MR 0330152 (48:8490)
  • [4] D. E. Dobbs, On going down for simple overrings, Proc. Amer. Math. Soc. 39 (1973), 515-519. MR 0417152 (54:5211)
  • [5] -, On going down for simple overrings. II, Comm. Algebra 1 (1974), 439-458. MR 0364225 (51:480)
  • [6] D. E. Dobbs and I. J. Papick, On going down for simple overrings. III, Proc. Amer. Math. Soc. 54 (1976), 35-38. MR 0417153 (54:5212)
  • [7] D. E. Dobbs, Ascent and descent of going-down rings for integral extensions (submitted).
  • [8] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 33 #1824. MR 0193606 (33:1824)
  • [9] D. Ferrand, Morphismes entiers universellement ouverts (manuscript).
  • [10] R. W. Gilmer, Jr. and W. J. Heinzer, Intersections of quotient rings of an integral domain, J. Math. Kyoto Univ. 7 (1967), 133-150. MR 36 #6397. MR 0223349 (36:6397)
  • [11] R. W. Gilmer, Jr., Multiplicative ideal theory, Queen's Papers in Pure and Appl. Math., no. 12, Queen's University, Kingston, Ont., 1968. MR 37 #5198. MR 0229624 (37:5198)
  • [12] -, The pseudo-radical of a commutative ring, Pacific J. Math. 19 (1966), 275-284. MR 34 #4294. MR 0204452 (34:4294)
  • [13] -, Prüfer-like conditions on the set of overrings of an integral domain, Conf. on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., vol. 311, Springer-Verlag, Berlin, 1973, pp. 90-102. MR 49 #5000. MR 0340245 (49:5000)
  • [14] R. W. Gilmer, Jr. and J. Huckaba, $ \Delta $-rings, J. Algebra 28 (1974), 414-432. MR 0427308 (55:342)
  • [15] B. Greenberg, Ph.D. Dissertation, Rutgers University, New Brunswick, N. J., 1973.
  • [16] A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique. IV, Inst. Hautes Études Sci. Publ. Math. No. 20 (1964). MR 30 #3885. MR 2334323 (2008g:46001)
  • [17] M. Hochster, Non-openness of loci in Noetherian rings, Duke Math. J. 40 (1973), 215-219. MR 47 #215. MR 0311653 (47:215)
  • [18] I. Kaplansky, Commutative rings, Allyn and Bacon, Boston, Mass., 1970. MR 40 #7234. MR 0254021 (40:7234)
  • [19] -, Topics in commutative ring theory, University of Chicago, Chicago, Ill. (mimeographed notes).
  • [20] W. J. Lewis, The spectrum of a ring as a partially ordered set, J. Algebra 25 (1973), 419-434. MR 47 #3361. MR 0314811 (47:3361)
  • [21] W. S. Massey, Algebraic topology: An introduction, Harcourt, Brace and World, New York, 1967. MR 35 #2271. MR 0211390 (35:2271)
  • [22] H. Matsumura, Commutative algebra, Benjamin, New York, 1970. MR 42 #1813. MR 0266911 (42:1813)
  • [23] S. McAdam, Going down in polynomial rings, Canad. J. Math. 23 (1971), 704-711. MR 43 #6202. MR 0280482 (43:6202)
  • [24] -, Going down, Duke Math. J. 39 (1972), 633-636. MR 47 #220. MR 0311658 (47:220)
  • [25] -, Going down and open extensions, Canad. J. Math. (to appear). MR 0357382 (50:9850)
  • [26] -, Simple going down, J. London Math. Soc. (to appear).
  • [27] M. Nagata, Local rings, Interscience Tracts in Pure and Appl. Math., no. 13, Interscience, New York, 1962. MR 27 #5790. MR 0155856 (27:5790)
  • [28] I. J. Papick, Classes of going-down domains, Ph. D. Dissertation, Rutgers University, New Brunswick, N. J., 1974.
  • [29] M. Raynaud, Anneaux locaux henséliens, Lecture Notes in Math., vol. 169, Springer-Verlag, Berlin and New York, 1970. MR 43 #3252. MR 0277519 (43:3252)
  • [30] F. Richman, Generalized quotient rings, Proc. Amer. Math. Soc. 16 (1965), 794-799. MR 31 #5880. MR 0181653 (31:5880)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0401745-0
Keywords: Going-down, treed, mated, open, propen, branch, G-domain, fiber, trunk, vertex
Article copyright: © Copyright 1976 American Mathematical Society

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