An application of valuation theory to rings of continuous real and complex-valued functions

Alling, Norman L.

doi:10.1090/S0002-9947-1963-0154886-0

An application of valuation theory to rings of continuous real and complex-valued functions
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by Norman L. Alling

Trans. Amer. Math. Soc. 109 (1963), 492-508

DOI: https://doi.org/10.1090/S0002-9947-1963-0154886-0

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References

Norman L. Alling, A characterization of Abelian $\eta _{\alpha }$-groups in terms of their natural valuation, Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 711–713. MR 175983, DOI 10.1073/pnas.47.5.711
Norman L. Alling, On the existence of real-closed fields that are $\eta _{\alpha }$-sets of power $\aleph _{\alpha }$, Trans. Amer. Math. Soc. 103 (1962), 341–352. MR 146089, DOI 10.1090/S0002-9947-1962-0146089-X

Über nicht-Archimedisch geordnete Körper

8

N. Bourbaki, Éléments de mathématique. XIV. Première partie: Les structures fondamentales de l’analyse. Livre II: Algèbre. Chapitre VI: Groupes et corps ordonnés. Chapitre VII: Modules sur les anneaux principaux, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1179, Hermann & Cie, Paris, 1952 (French). MR 0049861
A. H. Clifford, Completion of semi-continuous ordered commutative semigroups, Duke Math. J. 26 (1959), 41–59. MR 101276
A. H. Clifford, Totally ordered commutative semigroups, Bull. Amer. Math. Soc. 64 (1958), 305–316. MR 100641, DOI 10.1090/S0002-9904-1958-10221-9
Paul F. Conrad, Embedding theorems for abelian groups with valuations, Amer. J. Math. 75 (1953), 1–29. MR 53933, DOI 10.2307/2372611
Paul Conrad, On ordered division rings, Proc. Amer. Math. Soc. 5 (1954), 323–328. MR 61582, DOI 10.1090/S0002-9939-1954-0061582-7
P. Erdös, L. Gillman, and M. Henriksen, An isomorphism theorem for real-closed fields, Ann. of Math. (2) 61 (1955), 542–554. MR 69161, DOI 10.2307/1969812
L. Gillman, M. Henriksen, and M. Jerison, On a theorem of Gelfand and Kolmogoroff concerning maximal ideals in rings of continuous functions, Proc. Amer. Math. Soc. 5 (1954), 447–455. MR 66627, DOI 10.1090/S0002-9939-1954-0066627-6
Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199
Leonard Gillman and Carl W. Kohls, Convex and pseudoprime ideals in rings of continuous functions, Math. Z. 72 (1959/60), 399–409. MR 114115, DOI 10.1007/BF01162963
K. A. H. Gravett, Valued linear spaces, Quart. J. Math. Oxford Ser. (2) 6 (1955), 309–315. MR 87649, DOI 10.1093/qmath/6.1.309

Über die nichtarchimedischen Grössensysteme

116

Felix Hausdorff, Felix Hausdorff—gesammelte Werke. Band II, Springer-Verlag, Berlin, 2002 (German). “Grundzüge der Mengenlehre”. [“Foundations of set theory”]; Edited and with commentary by E. Brieskorn, S. D. Chatterji, M. Epple, U. Felgner, H. Herrlich, M. Hušek, V. Kanovei, P. Koepke, G. Preuß, W. Purkert and E. Scholz. MR 2229142
Melvin Henriksen and J. R. Isbell, On the continuity of the real roots of an algebraic equation, Proc. Amer. Math. Soc. 4 (1953), 431–434. MR 55664, DOI 10.1090/S0002-9939-1953-0055664-2
Edwin Hewitt, Rings of real-valued continuous functions. I, Trans. Amer. Math. Soc. 64 (1948), 45–99. MR 26239, DOI 10.1090/S0002-9947-1948-0026239-9
J. R. Isbell, More on the continuity of the real roots of an algebraic equation, Proc. Amer. Math. Soc. 5 (1954), 439. MR 62425, DOI 10.1090/S0002-9939-1954-0062425-8
H. Jerome Keisler, Ultraproducts and elementary classes, Nederl. Akad. Wetensch. Proc. Ser. A 64 = Indag. Math. 23 (1961), 477–495. MR 0140396
Irving Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303–321. MR 6161
C. W. Kohls, Ideals in rings of continuous functions, Fund. Math. 45 (1957), 28–50. MR 102731, DOI 10.4064/fm-45-1-28-50
Carl W. Kohls, Prime ideals in rings of continuous functions, Illinois J. Math. 2 (1958), 505–536. MR 102732

Allgemeine Bewertungstheorie

167

Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0120249