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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Primary ideals in rings of analytic functions


Author: R. Douglas Williams
Journal: Trans. Amer. Math. Soc. 177 (1973), 37-49
MSC: Primary 46J20; Secondary 13C05, 30A98
DOI: https://doi.org/10.1090/S0002-9947-1973-0320760-6
MathSciNet review: 0320760
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Abstract: Let A be the ring of all analytic functions on a connected, noncompact Riemann surface. We use the valuation theory of the ring A as developed by N. L. Alling to analyze the structure of the primary ideals of A. We characterize the upper and lower primary ideals of A and prove that every nonprime primary ideal of A is either an upper or a lower primary ideal. In addition we give some necessary and sufficient conditions for certain ideals of A to be intersections of primary ideals.


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

  • [A$ _{1}$] N. L. Alling, The valuation theory of meromorphic function fields over open Riemann surfaces, Acta Math. 110 (1963), 79-96. MR 28 #3992. MR 0160781 (28:3992)
  • [A$ _{2}$] -, The valuation theory of meromorphic function fields, Proc. Sympos. Pure Math., vol. 11, Amer. Math. Soc., Providence, R.I., 1968, pp. 8-29. MR 38 #4700. MR 0236404 (38:4700)
  • [F] H. Florack, Reguläre und meromorphe Funktionen auf nicht geschlossenen Riemannschen Flächen, Schr. Math. Inst. Univ. Munster No. 1 (1948). MR 12, 251. MR 0037362 (12:251i)
  • [GJ] L. Gillman and M. Jerison, Rings of continuous functions, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #6994. MR 0116199 (22:6994)
  • [H$ _{1}$] M. Henriksen, On the ideal structure of the ring of entire functions, Pacific J. Math. 2 (1952), 179-184. MR 13, 954. MR 0047928 (13:954f)
  • [H$ _{2}$] -, On the prime ideals of the ring of entire functions, Pacific J. Math. 3 (1953), 711-720. MR 15, 537. MR 0059479 (15:537c)
  • [K] C. W. Kohls, Primary ideals in rings, of continuous functions, Amer. Math. Monthly 71 (1964), 980-984. MR 30 #2332. MR 0172106 (30:2332)
  • [N] M. Nagata, Local rings, Pure and Appl. Math., no. 13, Interscience, New York, 1962. MR 27 #5790. MR 0155856 (27:5790)
  • [S] O. F. G. Schilling, The theory of valuations, Math. Surveys, no. 4, Amer. Math. Soc., Providence, R. I., 1950. MR 13, 315. MR 0043776 (13:315b)
  • [W] R. D. Williams, Intersections of primary ideals in rings of continuous functions, Canad. J. Math. 24 (1972), 502-519. MR 0295066 (45:4134)
  • [ZS] O. Zariski and P. Samuel, Commutative algebra. Vol. II, The University Series in Higher Math., Van Nostrand, Princeton, N. J., 1960. MR 22 #11006. MR 0120249 (22:11006)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0320760-6
Keywords: Rings of analytic functions, primary ideals, intersections of prime ideals, intersections of primary ideals, rings of continuous functions
Article copyright: © Copyright 1973 American Mathematical Society

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