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What is the minimum number of generators for $M_n(\mathbb Q)$, the set of $n \times n$ matrices over $\mathbb Q$, which will generate it as an algebra over $\mathbb Q$ ?

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    @Valeriya: Wow, five questions in half an hour! Take a breath!2011-09-30

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Here's an exercise from Jacobson's Basic Algebra I which says that the answer is 2. It's from the exercises in section 7.1 (page 410 in the second edition, page 391 in the first edition).

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    For me an easier generating set is E11 and the n-cycle permutation matrix P=A+E(n,1). { P^i*E11*P^j } = { Eij }, though the indexing looks better with 0-based matrices, I think.2011-09-30