Some formulas for the minimal number of generators of the direct sum of matrix rings
نویسنده
چکیده
We obtain an asymptotic upper bound for the minimal number of generators for a finite direct sum of matrix algebras with entries in a finite field. This produces an upper bound for a similar quantity for integer matrix rings. We obtain an exact formula for the minimal number of generators for a finite direct sum of 2-by-2 matrix algebras with entries in a finite field. As a consequence, we show that a direct sum of up to 16 copies of M2(Z) has 2 generators, i.e. every element of M2(Z) may be written as a noncommutative polynomial in these generators with coefficients in Z. (Therefore, the same is true if in the previous sentence Z is replaced with any ring with 1.) It also follows that the minimal number of generators for the ring M2(Z) is 3. 2000 MSC: 16S50, 15A30, 15A33, 15A36, 15A30, 16P90
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Some formulas for the smallest number of generators of finite direct sum of matrix algebras
We obtain an asymptotic upper bound for the smallest number of generators for a finite direct sum of matrix algebras with entries in a finite field. This produces an upper bound for a similar quantity for integer matrix rings. We also obtain an exact formula for the smallest number of generators for a finite direct sum of 2-by-2 matrix algebras with entries in a finite field and as a consequenc...
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