Prime ideals in polynomial rings over one-dimensional domains
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- by William Heinzer and Sylvia Wiegand PDF
- Trans. Amer. Math. Soc. 347 (1995), 639-650 Request permission
Abstract:
Let $R$ be a one-dimensional integral domain with only finitely many maximal ideals and let $x$ be an indeterminate over $R$. We study the prime spectrum of the polynomial ring $R[x]$ as a partially ordered set. In the case where $R$ is countable we classify $\operatorname {Spec} (R[x])$ in terms of splitting properties of the maximal ideals ${\mathbf {m}}$ of $R$ and the valuative dimension of ${R_{\mathbf {m}}}_{}$.References
- Robert Gilmer, Multiplicative ideal theory, Pure and Applied Mathematics, No. 12, Marcel Dekker, Inc., New York, 1972. MR 0427289
- William Heinzer, Noetherian intersections of integral domains. II, Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972), Lecture Notes in Math., Vol. 311, Springer, Berlin, 1973, pp. 107–119. MR 0340250
- William Heinzer and Sylvia Wiegand, Prime ideals in two-dimensional polynomial rings, Proc. Amer. Math. Soc. 107 (1989), no. 3, 577–586. MR 982402, DOI 10.1090/S0002-9939-1989-0982402-3
- William J. Heinzer, David Lantz, and Sylvia M. Wiegand, Prime ideals in birational extensions of polynomial rings, Commutative algebra: syzygies, multiplicities, and birational algebra (South Hadley, MA, 1992) Contemp. Math., vol. 159, Amer. Math. Soc., Providence, RI, 1994, pp. 73–93. MR 1266180, DOI 10.1090/conm/159/01505
- William Heinzer, David Lantz, and Sylvia Wiegand, Projective lines over one-dimensional semilocal domains and spectra of birational extensions, Algebraic geometry and its applications (West Lafayette, IN, 1990) Springer, New York, 1994, pp. 309–325. MR 1272038
- Paul Jaffard, Théorie de la dimension dans les anneaux de polynomes, Mémor. Sci. Math., Fasc. 146, Gauthier-Villars, Paris, 1960 (French). MR 0117256
- Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
- Stephen McAdam, Intersections of height $2$ primes, J. Algebra 49 (1977), no. 2, 315–321. MR 480481, DOI 10.1016/0021-8693(77)90242-3
- Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0155856
- Jack Ohm and R. L. Pendleton, Rings with noetherian spectrum, Duke Math. J. 35 (1968), 631–639. MR 229627
- A. Seidenberg, A note on the dimension theory of rings, Pacific J. Math. 3 (1953), 505–512. MR 54571, DOI 10.2140/pjm.1953.3.505
- Roger Wiegand, Homeomorphisms of affine surfaces over a finite field, J. London Math. Soc. (2) 18 (1978), no. 1, 28–32. MR 491732, DOI 10.1112/jlms/s2-18.1.28
- Roger Wiegand, The prime spectrum of a two-dimensional affine domain, J. Pure Appl. Algebra 40 (1986), no. 2, 209–214. MR 830322, DOI 10.1016/0022-4049(86)90041-1
Additional Information
- © Copyright 1995 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 347 (1995), 639-650
- MSC: Primary 13B25; Secondary 13A15
- DOI: https://doi.org/10.1090/S0002-9947-1995-1242087-5
- MathSciNet review: 1242087