George M Bergman

Abstract:  . Let k ? L be division rings, with [L : k ] right < , and let B k ? A L be right vector spaces. Conditions are established in terms of the k-codimension of B in A, respectively the k-dimension of B, for there to exist a nonzero element a?A such that aL?B = {0}, respectively an L-linear functional f on A such that f (B) = L . Some consequences and related open questions are discussed. Suppose k ? L are fields, or more generally, division rings, with [L : k ] right < . If A is a right L-vector-space and B a k-subspace of A , we may ask under what conditions there will exist a nonzero L-subspace of A having zero intersection with B. For the commutative algebraist, or the noncommutative algebraist to whom the only division rings of interest are finite-dimensional algebras over fields, the question is answered completely in 1, where it is shown that the obvious necessary condition for the existence of such a subspace, namely that the k-codimension of B in A be at least [L ...