Extensions of valuation rings in central simple algebras
HTML articles powered by AMS MathViewer
- by H.-H. Brungs and J. Gräter PDF
- Trans. Amer. Math. Soc. 317 (1990), 287-302 Request permission
Abstract:
Certain subrings $R$ of simple algebras $Q$, finite dimensional over their center $K$, are studied. These rings are called $Q$-valuation rings since they share many properties with commutative valuation rings. Let $V$ be a valuation ring of $K$, the center of $Q$, and let $\mathcal {R}$ be the set of $Q$-valuation rings $R$ in $Q$ with $R \cap K = V$, then $\left | \mathcal {R} \right | \geq 1$. This extension theorem, which does not hold if one considers only total valuation rings, was proved by N. I. Dubrovin. Here, first a somewhat different proof of this result is given and then information about the set $\mathcal {R}$ is obtained. Theorem. The elements in $\mathcal {R}$ are conjugate if $V$ has finite rank. Theorem. The elements in $\mathcal {R}$ are total valuation rings if $\mathcal {R}$ contains one total valuation ring. In this case $Q$ is a division ring. Theorem. $\mathcal {R}$ if $\mathcal {R}$ contains an invariant total valuation ring.References
- S. A. Amitsur, On central division algebras, Israel J. Math. 12 (1972), 408–420. MR 318216, DOI 10.1007/BF02764632
- Gorô Azumaya, On maximally central algebras, Nagoya Math. J. 2 (1951), 119–150. MR 40287
- H. H. Brungs, Rings with a distributive lattice of right ideals, J. Algebra 40 (1976), no. 2, 392–400. MR 409568, DOI 10.1016/0021-8693(76)90203-9
- Hans-Heinrich Brungs and Günter Törner, Extensions of chain rings, Math. Z. 185 (1984), no. 1, 93–104. MR 724046, DOI 10.1007/BF01214974
- H.-H. Brungs and Joachim Gräter, Valuation rings in finite-dimensional division algebras, J. Algebra 120 (1989), no. 1, 90–99. MR 977862, DOI 10.1016/0021-8693(89)90190-7
- P. M. Cohn, On the embedding of rings in skew fields, Proc. London Math. Soc. (3) 11 (1961), 511–530. MR 136632, DOI 10.1112/plms/s3-11.1.511
- P. M. Cohn and M. Mahdavi-Hezavehi, Extensions of valuations on skew fields, Ring theory, Antwerp 1980 (Proc. Conf., Univ. Antwerp, Antwerp, 1980), Lecture Notes in Math., vol. 825, Springer, Berlin, 1980, pp. 28–41. MR 590783
- P. M. Cohn, On extending valuations in division algebras, Studia Sci. Math. Hungar. 16 (1981), no. 1-2, 65–70. MR 703642
- N. I. Dubrovin, Noncommutative valuation rings, Trudy Moskov. Mat. Obshch. 45 (1982), 265–280 (Russian). MR 704633
- N. I. Dubrovin, Noncommutative valuation rings in simple finite-dimensional algebras over a field, Mat. Sb. (N.S.) 123(165) (1984), no. 4, 496–509 (Russian). MR 740675
- Otto Endler, Valuation theory, Universitext, Springer-Verlag, New York-Heidelberg, 1972. To the memory of Wolfgang Krull (26 August 1899–12 April 1971). MR 0357379
- Alfred W. Goldie, The structure of Noetherian rings, Lectures on rings and modules (Tulane Univ. Ring and Operator Theory Year, 1970-1971, Vol. I), Lecture Notes in Math., Vol. 246, Springer, Berlin, 1972, pp. 213–321. MR 0393118
- Joachim Gräter, Zur Theorie nicht kommutativer Prüferringe, Arch. Math. (Basel) 41 (1983), no. 1, 30–36 (German). MR 713664, DOI 10.1007/BF01193819
- J. Gräter, Über Bewertungen endlich dimensionaler Divisionsalgebren, Results Math. 7 (1984), no. 1, 54–57 (German). MR 758767, DOI 10.1007/BF03322487
- Bill Jacob and Adrian R. Wadsworth, A new construction of noncrossed product algebras, Trans. Amer. Math. Soc. 293 (1986), no. 2, 693–721. MR 816320, DOI 10.1090/S0002-9947-1986-0816320-X
- Wilhelm Klingenberg, Projektive und affine Ebenen mit Nachbarelementen, Math. Z. 60 (1954), 384–406 (German). MR 65938, DOI 10.1007/BF01187385 W. Krull, Allgemeine Bewertungstheorie, J. Reine Angew. Math. 167 (1932), 160-196.
- James Kuzmanovich, Localizations of Dedekind prime rings, J. Algebra 21 (1972), 378–393. MR 311698, DOI 10.1016/0021-8693(72)90002-6
- A. I. Lichtman, PI-subrings and algebraic elements in enveloping algebras and their fields of fractions, J. Algebra 121 (1989), no. 1, 139–154. MR 992321, DOI 10.1016/0021-8693(89)90090-2
- Karl Mathiak, Bewertungen nicht kommutativer Körper, J. Algebra 48 (1977), no. 2, 217–235 (German). MR 485810, DOI 10.1016/0021-8693(77)90304-0
- Karl Mathiak, Valuations of skew fields and projective Hjelmslev spaces, Lecture Notes in Mathematics, vol. 1175, Springer-Verlag, Berlin, 1986. MR 835210, DOI 10.1007/BFb0074653
- B. H. Neumann, On ordered division rings, Trans. Amer. Math. Soc. 66 (1949), 202–252. MR 32593, DOI 10.1090/S0002-9947-1949-0032593-5
- B. L. Osofsky, Noncommutative rings whose cyclic modules have cyclic injective hulls, Pacific J. Math. 25 (1968), 331–340. MR 231858
- I. Reiner, Maximal orders, London Mathematical Society Monographs, No. 5, Academic Press [Harcourt Brace Jovanovich, Publishers], London-New York, 1975. MR 0393100
- O. F. G. Schilling, Noncommutative valuations, Bull. Amer. Math. Soc. 51 (1945), 297–304. MR 11684, DOI 10.1090/S0002-9904-1945-08339-6
- O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys, No. 4, American Mathematical Society, New York, N. Y., 1950. MR 0043776 M. Schràder, Angeordnete Schiefkàrper mit natürlicher Bewertung vom Rang 1, Dissertation, Münster, 1985.
- Adrian R. Wadsworth, Extending valuations to finite-dimensional division algebras, Proc. Amer. Math. Soc. 98 (1986), no. 1, 20–22. MR 848866, DOI 10.1090/S0002-9939-1986-0848866-8
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 317 (1990), 287-302
- MSC: Primary 16A39
- DOI: https://doi.org/10.1090/S0002-9947-1990-0946216-5
- MathSciNet review: 946216