Geodesic
Maximal orders in a finite-dimensional central simple algebra over a valuation ring of height 1
Sbornik. Mathematics, Tome 36 (1980) no. 4, pp. 483-493 
N. I. Dubrovin. Maximal orders in a finite-dimensional central simple algebra over a valuation ring of height 1. Sbornik. Mathematics, Tome 36 (1980) no. 4, pp. 483-493. http://geodesic.mathdoc.fr/item/SM_1980_36_4_a2/
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Voir la notice de l'article provenant de la source Math-Net.Ru

This paper consists of two sections. In § maximal, almost maximal, and complete valuation rings are characterized in terms of the decomposability of torsion-free modules of rank 2. In § 2 an attempt is made to describe the maximal $V$-orders in the matrix ring $K_n$, where $V$ is a valuation ring of height 1 in the field $K$. Also, § 2 contains a generalization to a matrix algebra over a field of the well-known fact that a maximal subring of a field is either a field or a valuation ring of height 1. Bibliography: 9 titles.

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[2] E. Matlis, “Delomposable modules”, Trans. Amer. Math. Soc., 125 (1966), 147–179 | DOI | MR | Zbl

[3] I. Kaplansky, “Elementary divisors and modules”, Trans. Amer. Math. Soc., 66 (1949), 464–491 | DOI | MR | Zbl

[4] D. T. Gill, “Almost maximal valuation rings”, J. London Math. Soc., Ser. 2, 4:1 (1971), 140–146 | DOI | MR | Zbl

[5] G. B. Klatt, L. S. Levy, “Pre-self-injective rings”, Trans. Amer. Math. Soc., 137 (1968), 407–419 | DOI | MR

[6] O. F. G. Schilling, The theory of valuations, Math. Surveyes, 4, Amer. Math. Soc., Providence, RI, 1950, MR/3, 316 | MR

[7] J. J. Kelleher, “Totally ordered sets of ideals in rings of analytic functions”, J. reine und angew. Math., 278–279 (1975), 118–156 | MR

[8] H. Burbaki, Kommutativnaya algebra, izd-vo “Mir”, Moskva, 1971 | MR

[9] Yu. P. Razmyslov, “Ob odnoi probleme Kaplanskogo”, Izv. AN SSSR, seriya matem., 37:3 (1973), 483–501 | MR | Zbl