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Integral Dubrovin valuation rings


Authors: Patrick J. Morandi and Adrian R. Wadsworth
Journal: Trans. Amer. Math. Soc. 315 (1989), 623-640
MSC: Primary 16W60; Secondary 16K20, 16S35
DOI: https://doi.org/10.1090/S0002-9947-1989-0986696-4
MathSciNet review: 986696
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Abstract: In the preceding paper, Dubrovin valuation rings integral over their centers in central simple algebras were characterized by value functions. Here, these value functions are used to give a method for extending integral Dubrovin valuation rings in generalized crossed product algebras. Several applications of this extension theorem are given, including new and more natural proofs of some theorems on valued division algebras over Henselian fields.


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  • [BG] H. H. Brungs and J. Gräter, Extensions of valuation rings in central simple algebras, Trans. Amer. Math. Soc. (to appear). MR 946216 (90d:16023)
  • [Dr] P. Draxl, Ostrowski's theorem for Henselian valued skew fields, J. Reine Angew. Math. 354 (1984), 213-218. MR 767581 (86g:12008)
  • [D$ _{1}$] N. I. Dubrovin, Noncommutative valuation rings, Trudy Moskov. Mat. Obshch. 45 (1982), 265-280; English transl., Trans. Moscow Math. Soc. 45 (1984), 273-287. MR 704633 (85d:16002)
  • [D$ _{2}$] N. I. Dubrovin, Noncommutative valuation rings in simple finite-dimensional algebras over a field, Mat. Sb. 123 (165) (1984), 496-509; English transl., Math. USSR-Sb. 51 (1985), 493-505. MR 740675 (85j:16020)
  • [JW] B. Jacob and A. Wadsworth, Division algebras over Henselian fields, J. Algebra (to appear). MR 1031915 (91d:12006)
  • [J$ _{1}$] W. Jehne, Idelklassenfaktorensysteme und verallgemeinerte Theorie der verschränkten Produkte, Abh. Math. Sem. Hamburg 18 (1952), 70-98. MR 0053160 (14:729b)
  • [J$ _{2}$] W. Jehne, Separable adel algebras and a presentation of Weil groups, J. Reine Angew. Math. 375/376 (1987), 211-237. MR 882298 (88j:11082)
  • [KY] V. V. Kursov and V. I. Yanchevskii, Crossed products over simple algebras and their automorphism groups, Akad. Nauk Belo. Inst. Math., preprint # 35, Minsk, 1987 (Russian).
  • [M$ _{1}$] P. Morandi, The Henselization of a valued division algebra, J. Algebra 122 (1989), 232-243. MR 994945 (90h:12007)
  • [M$ _{2}$] -, Value functions on central simple algebras, Trans. Amer. Math. Soc. 315 (1989), 605-622. MR 986697 (90e:16007)
  • [R] I. Reiner, Maximal orders, Academic Press, London, 1975. MR 1972204 (2004c:16026)
  • [T] J.-P. Tignol, Generalized crossed products, Séminaire Mathématique (nouvelle série), Rapport No. 106, Université Catholique de Louvain, Louvain-la-Neuve, Belgium, 1987.
  • [W$ _{1}$] A. Wadsworth, Dubrovin valuation rings and Henselization, Math. Ann. 283 (1989), 301-328. MR 980600 (90f:16009)
  • [W$ _{2}$] -, The residue division ring of a valued division algebra, Bull. Soc. Math. Belg. Ser. A 40 (1988), 307-322. MR 976644 (90b:16028)

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DOI: https://doi.org/10.1090/S0002-9947-1989-0986696-4
Article copyright: © Copyright 1989 American Mathematical Society

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