Let F be a field, and let A be a finite-dimensional F-algebra. Write d = dim(F) A, and let e be the largest degree of the minimal polynomial for any a is an element of A. Define the function f(d, e) = e root 2d/(e - 1) + 1/4 + e/2 - 2. We prove that, if S is any finite generating set for A as an F-algebra, the words in S of length less than f(d, e) span A as an F-vector space. In the special case of n-by-n matrices, this bound becomes f(n(2), n) = n root 2n(2)/(n - 1) + 1/4 + n/2 - 2 is an element of O(n(3/2)). This is a substantial improvement over previous bounds, which have all been O(n(2)). We also prove that, for particular sets S of matrices, the bound can be sharpened to one that is linear in n. As an application of these results, we reprove a theorem of Small, Stafford, and Warfield about semiprime affine F-algebras. (C) 1997 Academic Press. [References: 7]
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