Showing papers by "Zinovy Reichstein published in 1996"

On a question of Makar-Limanov

Abstract: Let K be an uncountable field, let K C F be a field extension, and let A be an associative K-algebra. We show that if F OK A contains a non-commutative free algebra, then so does A. Throughout this note K will be a field and Ko will be the prime subfield of K. Let A be an associative K-algebra. By this we mean, in particular, that K is contained in the center of A and IK = 1A We would like to know whether or not A contains (a) a free semi-group and (b) a free K0-algebra. Both of these free objects are presumed to be non-commutative on two generators. For a more detailed discussion of free subobjects of associative algebras we refer the reader to [LI], [L2], [LM] and [K]. The following result was conjectured by Makar-Limanov. Conjecture 1. Let D be a skew field, let K be a subfield of its center, and let F be a field extension of K. If F OK D contains a free Ko-algebra, then so does D. In this note we prove this conjecture under the additional assumption that K is an uncountable field. Our main result is the following theorem. Theorem 1. Let K be an uncountable field, A an associative K-algebra, and F a field extension of K. Denote the common prime field of K and F by Ko. (a) If F OK A contains a copy of the free semi-group, then so does A. (b) If F OK A contains a copy of the free Ko-algebra, then so does A. We remark that by [LM, Lemma 1] elements x, y E A generate a free subalgebra over Ko if and only if they generate a free subalgebra over K. Note that since we are assuming IA = IK, the argument of [LM, Lemma 1] goes through even if A is not a domain. Our proof of Theorem 1 is based on a general position argument. The condition that a pair of elements generates a free object in A is given by a countable number of polynomial inequalities; see Lemmas 2 and 3. Thus over an uncountable field the set of all such pairs behaves very much like an open set in the Zariski topology. In particular, we can prove the existence of a K-point by exhibiting an F-point. We now make these ideas precise. Lemma 1. Let K be an uncountable field and let X1, X2, ... be a countable number of Zariski closed subsets of K'. If U,1 Xi = K'm then Xi = K' for some i > 1. Received by the editors April 11, 1994 and, in revised form, June 24, 1994. 1991 Mathematics Subject Classification. Primary 16S10; Secondary 20M05. (?)1996 American Mathematical Society

Rational central simple algebras

Abstract: We introduce a notion of rationality (called toroidal or t-rationality) for central simple algebras which extends Demazure's characterization of rational algebraic varieties via torus actions. We prove a structure theorem for t-rational central simple algebras and study the interplay among t-rationality, crossed products and rationality of the center in the setting of universal division algebras.