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An Upper Bound for the Length of a Finite-Dimensional Algebra by Christopher J Pappacena is a Mathematics article available to read on EtoBox.

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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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Author

Christopher J Pappacena

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0021-8693)

Published

1997

Language

EN

Field

Mathematics (Physical Sciences)