Rational operators of the space of formal series

The main result of this paper is the following theorem: the group ring of the universal covering of the group SL(2, ℝ) is embeddable in a skew field with valuation in the sense of Mathiak and the valuation ring is an exceptional chain order in the skew field , 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 satisfies sufficient conditions for the embeddability of the group ring of a left ordered group in a skew field. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 3, pp. 9–53, 2006.