An Upper Bound for the Length of a Finite-Dimensional Algebra Original Research Article

LetFbe a field, and letAbe a finite-dimensionalF-algebra. Writed = dimF A, and letebe the largest degree of the minimal polynomial for anya set membership, variant A. Define the functionimage. We prove that, ifSis any finite generating set forAas anF-algebra, the words inSof length less thanf(d, e) spanAas anF-vector space. In the special case ofn-by-nmatrices, this bound becomesimage. This is a substantial improvement over previous bounds, which have all beenO(n2). We also prove that, for particular setsSof matrices, the bound can be sharpened to one that is linear inn. As an application of these results, we reprove a theorem of Small, Stafford, and Warfield about semiprime affineF-algebras.