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On pairs of matrices generating matrix rings and their presentations. (English summary) J. Algebra 310 (2007), no. 1, 15–40. In this excellent paper, which contains a large number of very interesting results and some highly nontrivial proofs, the authors study the complete matrix ring Mn(Z) of n × n matrices (n ≥ 2) with entries from the integers Z. The properties of Mn(Z) are based entirely on the presentation by the set {Eij: 1 ≤ i, j ≤ n} of n 2 matrix units, subject to the relations EijEkl = δjkEil. This set of n 2 generators can be reduced. It is also known that Mn(Z) is generated by the matrices X: = E2,1 + E3,2 + · · · + En,n−1 + E1,n and Y: = E11. The authors use X and Y to construct a number of presentations of Mn(Z) with two generators and finitely many relations, and they investigate the interdependence between the relations in these presentations. Amongst a plethora of other results, they also show that certain direct sums of matrix rings over Z and Q (the rational numbers) have presentations with two generators and finitely many relations

Year: 1977

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