Abstract
The main result of this paper is the following theorem: the group ring of the universal covering \(\mathbb{G}\) of the group SL(2, ℝ) is embeddable in a skew field \(\mathbb{R}\) with valuation in the sense of Mathiak and the valuation ring is an exceptional chain order in the skew field \(\mathbb{R}\), i.e., there exists a prime ideal that is not completely prime. In this ring, every divisorial right fractional ideal is principal, and the linearly ordered set of all divisorial fractional right ideals is isomorphic to the real line. This theorem is a consequence of the fact that the universal covering group \(\mathbb{G}\) satisfies sufficient conditions for the embeddability of the group ring of a left ordered group in a skew field.
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References
N. I. Dubrovin, “Invertibility of the group ring of a right-ordered group over a skew field,” Math. Notes, 42, No. 3/4, 781–786 (1987).
N. I. Dubrovin, “Rational closures of group rings of left-ordered groups,” Russian Acad. Sci. Sb. Math., 79, No. 2, 231–263 (1994).
N. I. Dubrovin, The Rational Closure of Group Rings of Left-Ordered Groups, SM-DU-254, Duisburg (1994).
N. I. Dubrovin, “Formal sums and power series over a group,” Russian Acad. Sci. Sb. Math., 191, No. 7, 955–971 (2000).
N. I. Dubrovin and H. H. Brungs, “A classification and examples of rank one chain domains,” Trans. Amer. Math. Soc., 335, No. 7, 2733–2753 (2003).
N. I. Dubrovin, J. Gräter, and T. Hanke, “Complexity of elements in rings,” Algebr. Represent. Theory., 6, 33–45 (2003).
T. V. Dubrovina and N. I. Dubrovin, “Cones in groups,” Russian Acad. Sci. Sb. Math., 187, No. 7, 1005–1019 (1996).
T. V. Dubrovina and N. I. Dubrovin, “Roots in the universal covering group of the unimodular (2 × 2)-matrix group,” Fundam. Prikl. Mat., 6, No. 3, 757–776 (2000).
T. V. Dubrovina and N. I. Dubrovin, “Topological linear spaces of formal sums,” Mat. Stud., 21, No. 2, 209–220 (2004).
K. Mathiak, “Zur Bewertungstheorie nicht kommutativer Körper,” J. Algebra, 73, No. 2, 586–560 (1981).
M. M. Postnikov, Lectures in Geometry. Semester V: Lie Groups and Lie Algebras [in Russian], Nauka, Moscow (1982).
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Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 9–53, 2006.
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Dubrovin, N.I. Rational operators of the space of formal series. J Math Sci 149, 1191–1223 (2008). https://doi.org/10.1007/s10958-008-0059-3
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DOI: https://doi.org/10.1007/s10958-008-0059-3