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Dubrovin valuation rings and Henselization

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References

  • [B] Bourbaki, N.: Algèbre commutative. Che. 6, valuations. Paris: Hermann 1964

    Google Scholar 

  • [BG1] Brungs, H.H., Gräter, J.: Valuation rings in finite dimensional division algebras. J. Algebra (to appear)

  • [BG2] Brungs, H.H., Gräter, J.: Extensions of valuation rings in central simple algebras. Trans. Am. Math. Soc. (to appear)

  • [C] Cohn, P.M.: Algebra, Vol. 2. London: Wiley 1977

    Google Scholar 

  • [DI] Demeyer, F., Ingraham, E.: Separable algebras over commutative rings. (Lectures Notes in Math., Vol. 181). Berlin Heidelberg New York: Springer 1971

    Google Scholar 

  • [DR] Draxl, P.: Ostrowski's theorem for Henselian valued skew fields. J. Reine Angew. Math.354, 213–218 (1984)

    Google Scholar 

  • [DK] Draxl, P., Kneser, M. (eds.):SK 1 von Schiefkörpern. (Lecture Notes in Math., Vol. 778). Berlin: Springer 1980

    Google Scholar 

  • [D1] Dubrovin, N.I.: Noncommutative valuation rings. Tr. Mosk. Mat. O.-va.,45, 265–280 (1982). English trans: Trans. Mosc. Math. Soc.45, 273–287 (1984)

    Google Scholar 

  • [D2] Dubrovin, N.I.: Noncommutative valuation rings in simple finite-dimensional algebras over a field. Mat. Sb.123, 496–509 (1984); English trans: Math. USSR Sb.51, 493–505 (1985)

    Google Scholar 

  • [E] Endler, O.: Valuation theory. New York: Springer 1972

    Google Scholar 

  • [Er1] Ershov, Yu.L.: Henselian valuations of division rings and the group SK1. Mat. Sb.117, 60–68 (1982); English transl.: Math. USSR Sb.45, 63–71 (1983)

    Google Scholar 

  • [Er2] Ershov, Yu.L.: Valued division rings, pp. 53–55, in Fifth All Union Symposium on the Theory of Rings, Algebras, and Modules, Akad. Nauk SSSR Sibirsk. Otdel., Inst. Mat., Novosibirsk, 1982 (in Russian)

    Google Scholar 

  • [F] Formanek, E.: Noetherian PI-Rings. Commun. Algebra1, 79–86 (1974)

    Google Scholar 

  • [G] Gräter, J.: Valuations on finite-dimensional division algebras and their value groups. Arch. Math.51, 128–140 (1988)

    Google Scholar 

  • [H] Hua, L.K.: A note on the total matrix ring over a non-commutative field. Ann. Soc. Math. Pol.25, 188–198 (1952)

    Google Scholar 

  • [JW] Jacob, B., Wadsworth, A.: Division algebras over Henselian fields. J. Algebra (to appear)

  • [K] Kasch, F.: Invariante Untermoduln des Endomorphismenringes eines Vektorraums. Arch. Math.4, 182–190 (1953)

    Google Scholar 

  • [Ma] Mathiak, K.: Valuations of skew fields and projective Hjelmslev spaces. (Lecture Notes in Math., Vol. 1175). Berlin Heidelberg New York: Springer 1986

    Google Scholar 

  • [M1] Morandi, P.: The Henselization of a valued division algebra. J. Algebra (to appear)

  • [M2] Morandi, P.: Valuation rings in division rings and central simple algebras, doctoral dissertation, Univ. of Calif. at San Diego, 1988

  • [OS] Ojanguren, M., Sridharan, R.: Cancellation of Azumaya algebras. J. Algebra18, 501–505 (1971)

    Google Scholar 

  • [Re] Reiner, I.: Maximal orders. London: Academic Press 1975

    Google Scholar 

  • [R1] Ribenboim, P.: Théorie des valuations. Montréal: Presses Univ. Montréal 1968

    Google Scholar 

  • [R2] Ribenboim, P.: Equivalent forms of Hensel's lemma. Expo. Math.3, 3–24 (1985)

    Google Scholar 

  • [Ro] Rosenberg, A.: The Cartan-Brauer-Hua theorem for matrix and local matrix rings. Proc. Am. Math. Soc.7, 891–898 (1956)

    Google Scholar 

  • [Rw] Rowen, L.H.: Polynomial identities in ring theory. New York: Academic Press 1980

    Google Scholar 

  • [Sa] Saltman, D.: The Brauer group and the center of generic matrices. J. Algebra97, 53–67 (1985)

    Google Scholar 

  • [S] Schilling, O.F.G.: The theory of valuations. Math. Surveys, No. 4, Am. Math. Soc., Providence, R.I., 1950

    Google Scholar 

  • [W1] Wadsworth, A.R.: Extending valuations to finite dimensional division algebras. Proc. Am. Math. Soc.98, 20–22 (1986)

    Google Scholar 

  • [W2] Wadsworth, A.R.: Dubrovin valuation rings. pp. 359–374. In Perspectives in ring theory, F. van Oystaeyen, L. LeBruyn (eds.) NATO ASI Series, Series C. Vol. 233. Dordrecht: Kluwer 1988

    Google Scholar 

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Supported in part by the National Science Foundation

Some of the research for this paper was carried out while the author was visiting the Wilhelms-Westfälische Universität of Münster, West Germany and the Université Catholique de Louvain of Louvain-la-Neuve, Belgium. The author would like to thank the mathematicians at both universities for their kind hospitality

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Wadsworth, A.R. Dubrovin valuation rings and Henselization. Math. Ann. 283, 301–328 (1989). https://doi.org/10.1007/BF01446437

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