Abstract
We prove that every separable algebra over an infinite field F admits a presentation with 2 generators and finitely many relations. In particular, this is true for finite direct sums of matrix algebras over F and for group algebras FG, where G is a finite group such that the order of G is invertible in F. We illustrate the usefulness of such presentations by using them to find a polynomial criterion to decide when 2 ordered pairs of 2 × 2 matrices (A, B) and (A′, B′) with entries in a commutative ring R are automorphically conjugate over the matrix algebra M 2(R), under an additional assumption that both pairs generate M 2(R) as an R-algebra.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Mazur, M., Petrenko, B.V. Separable algebras over infinite fields are 2-generated and finitely presented. Arch. Math. 93, 521–529 (2009). https://doi.org/10.1007/s00013-009-0058-8
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DOI: https://doi.org/10.1007/s00013-009-0058-8