Henriksen’s contributions to residue class rings of analytic and entire functions. (English) Zbl 1234.30025
Summary: The author surveys, summarizes and generalizes results of Golasiński and Henriksen, and of others, concerning certain residue class rings.
Let \(\mathcal A(\mathbb R)\) denote the ring of analytic functions over reals \(\mathbb R\) and \(\mathcal E(\mathbb K)\) the ring of entire functions over \(\mathbb R\) or complex numbers \(\mathbb C\). It is shown that if \(m\) is a maximal ideal of \(\mathcal A(\mathbb R)\), then \(\mathcal A(\mathbb R)/m\) is isomorphic either to the reals or a real-closed field that is \(\eta _{1}\)-set, while if \(m\) is a maximal ideal of \(\mathcal E(\mathbb K)\), then \(\mathcal E(\mathbb K)/m\) is isomorphic to one of these latter two fields or to complex numbers.
Let \(\mathcal A(\mathbb R)\) denote the ring of analytic functions over reals \(\mathbb R\) and \(\mathcal E(\mathbb K)\) the ring of entire functions over \(\mathbb R\) or complex numbers \(\mathbb C\). It is shown that if \(m\) is a maximal ideal of \(\mathcal A(\mathbb R)\), then \(\mathcal A(\mathbb R)/m\) is isomorphic either to the reals or a real-closed field that is \(\eta _{1}\)-set, while if \(m\) is a maximal ideal of \(\mathcal E(\mathbb K)\), then \(\mathcal E(\mathbb K)/m\) is isomorphic to one of these latter two fields or to complex numbers.
MSC:
| 30D20 | Entire functions of one complex variable (general theory) |
| 26E05 | Real-analytic functions |
| 13A15 | Ideals and multiplicative ideal theory in commutative rings |
| 13F25 | Formal power series rings |
Keywords:
adequate ring; algebraically (real) closed field; analytic (entire) function; Bezout domain; filter; integral extension; Krull dimension; maximal (prime) idealReferences:
| [1] | Bochnak, J.; Coste, M.; Roy, M.-F., Real Algebraic Geometry, Erg. der Math., vol. 36 (1998), Springer-Verlag: Springer-Verlag Berlin-Heidelberg-New York · Zbl 0633.14016 |
| [2] | Brewer, J. M.; Bunce, J. W.; van Vleck, F. S., Linear Systems over Commutative Rings, Lecture Notes in Pure and Applied Mathematics, vol. 104 (1986), Marcel Dekker: Marcel Dekker Inc., New York · Zbl 0607.13001 |
| [3] | Dow, A., On ultrapowers of Boolean algebras, Proceedings of the Topology Conference. Proceedings of the Topology Conference, Auburn, AL, 1984. Proceedings of the Topology Conference. Proceedings of the Topology Conference, Auburn, AL, 1984, Topology Proc., 9, 2, 269-291 (1984) · Zbl 0598.03040 |
| [4] | Erdös, P.; Gillman, L.; Henriksen, M., An isomorphism theorem for real-closed fields, Ann. of Math. (2), 61, 542-554 (1955) · Zbl 0065.02305 |
| [5] | Germay, R. H.J., Extension dʼun théorème de E. Picard relatif aux produits indéfinis de facteurs primaires, Bull. Roy. Sci. Liège, 17, 138-143 (1948) · Zbl 0034.04801 |
| [6] | Gillman, L.; Jerison, M., Rings of Continuous Functions (1960), D. van Nostrand Company, Inc.: D. van Nostrand Company, Inc. Princeton, New Jersey · Zbl 0093.30001 |
| [7] | Golasiński, M.; Henriksen, M., Residue class rings of real-analytic and entire functions, Colloq. Math., 104, 1, 85-97 (2006) · Zbl 1093.26027 |
| [8] | Golasiński, M., A simple embedding of meromorphic functions into complex numbers, Algebras Groups Geom., 22, 2, 147-149 (2005) · Zbl 1079.30034 |
| [9] | Helmer, O., Divisibility properties of integral functions, Duke Math. J., 6, 345-356 (1940) · JFM 66.0105.02 |
| [10] | Helmer, O., The elementary divisor theorem for certain rings without chain conditions, Bull. Amer. Math. Soc., 49, 225-236 (1943) · Zbl 0060.07606 |
| [11] | Henriksen, M., On the ideal structure of the ring of entire functions, Pacific J. Math., 2, 179-184 (1952) · Zbl 0048.09001 |
| [12] | Henriksen, M., On the prime ideals of the ring of entire functions, Pacific J. Math., 3, 711-720 (1953) · Zbl 0052.34101 |
| [13] | Henriksen, M., Some remarks on elementary divisor rings, II, Michigan Math. J., 3, 159-163 (1955/56) · Zbl 0073.02301 |
| [14] | Jensen, C. U., Some curiosities of rings of analytic functions, J. Pure Appl. Algebra, 38, 277-283 (1985) · Zbl 0584.30046 |
| [15] | Larsen, M.; Lewis, W.; Shores, T., Elementary divisor rings and finitely generated modules, Trans. Amer. Math. Soc., 187, 231-247 (1974) · Zbl 0283.13002 |
| [16] | Rubel, Lee A., Sums of squares of real entire functions of one variable, (Proceedings of the Conference on Complex Analysis. Proceedings of the Conference on Complex Analysis, Tianjin, 1992. Proceedings of the Conference on Complex Analysis. Proceedings of the Conference on Complex Analysis, Tianjin, 1992, Conf. Proc. Lecture Notes Anal., vol. I (1994), Internat. Press: Internat. Press Cambridge, MA), 180-187 · Zbl 0838.30025 |
| [17] | Schilling, O. F.G., Ideal theory on open Riemann surfaces, Bull. Amer. Math. Soc., 52, 945-963 (1946) · Zbl 0063.06792 |
| [18] | Steinitz, E., Algebraische Theorie der Körper (1930), Walter de Gruyter and Co.: Walter de Gruyter and Co. Berlin · JFM 56.0137.02 |
| [19] | Wedderburn, J. M.H., On matrices whose coefficients are functions of a single variable, Trans. Amer. Math. Soc., 16, 328-332 (1915) · JFM 45.0264.04 |
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