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On Pairs of Matrices Generating Matrix Rings and Their Presentations by B.V. Petrenko; S.N. Sidki is a Mathematics article available to read on EtoBox.

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Let M n (Z) be the ring of n-by-n matrices with integral entries, and n 2. This paper studies the set G n (Z) of pairs (A, B) ∈ M n (Z) 2 generating M n (Z) as a ring. We use several presentations of M n (Z) with generators X = n i=1 E i+1,i and Y = E 11 to obtain the following consequences. (1) Let k 1. The following rings have presentations with 2 generators and finitely many relations: (a) (Z), where n 1 , . . . , n k 2, and the same n i is repeated no more than three times. (2) Let D be a commutative domain of sufficiently large characteristic over which every finitely generated projective module is free. We use 4 relations for X and Y to describe all representations of the ring M n (D) into M m (D) for m n. (3) We obtain information about the asymptotic density of G n (F ) in M n (F ) 2 over different fields, and over the integers.

Who reads On Pairs of Matrices Generating Matrix Rings and Their Presentations?

Author

B.V. Petrenko; S.N. Sidki

Publisher

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

Published

2007

Language

EN

Field

Mathematics (Physical Sciences)