Matrix Rings and Linear Groups Over a Field of Fractions of Enveloping Algebras and Group Rings, I
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF ALGEBRA 88, 1-37 (1984) Matrix Rings and Linear Groups over a Field of Fractions of Enveloping Algebras and Group Rings, I A. I. LICHTMAN Department of Mathematics, Ben Gurion University of the Negev, Beer-Sheva, Israel, and Department of Mathematics, The University of Texas at Austin, Austin, Texas * Communicated by I. N. Herstein Received September 20, 1982 1. INTRODUCTION 1.1. Let H be a Lie algebra over a field K, U(H) be its universal envelope. It is well known that U(H) is a domain (see [ 1, Section V.31) and the theorem of Cohn (see [2]) states that U(H) can be imbedded in a field.’ When H is locally finite U(H) has even a field of fractions (see [ 1, Theorem V.3.61). The class of Lie algebras whose universal envelope has a field of fractions is however .much wider than the class of locally finite Lie algebras; it follows from Proposition 4.1 of the article that it contains, for instance, all the locally soluble-by-locally finite algebras. This gives a negative answer of the question in [2], page 528. Let H satisfy any one of the following three conditions: (i) U(H) is an Ore ring and fi ,“=, Hm = 0. (ii) H is soluble of class 2. (iii) Z-I is finite dimensional over K. Let D be the field of fractions of U(H) and D, (n > 1) be the matrix ring over D.
[Show full text]