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

✍ Scribed by Christopher J Pappacena

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 180 KB
Volume: 197
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1997.7140

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✦ Synopsis

Let F be a field, and let A be a finite-dimensional F-algebra. Write d s dim A, F and let e be the largest degree of the minimal polynomial for any a g A. Define Ž .

' the function f d, e s e 2dr e y 1 q 1r4 q er2 y 2. We prove that, if S is Ž .

any finite generating set for A as an F-algebra, the words in S of length less than Ž . f d, e span A as an F-vector space. In the special case of n-by-n matrices, this

' bound becomes f n , n s n 2n r n y 1 q 1r4 q nr2 y 2 g O n . This is Ž .

Ž 2 . a substantial improvement over previous bounds, which have all been O n . 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.

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