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Commutators and companion matrices over rings of stable rank 1. (English) Zbl 0713.15003
Let A be a ring of stable rank 1, i.e. if a,b\(\in A\) and \(Aa+Ab=A\), then there is some \(c\in A\) such that \(A(a+cb)=A\). As the main result the authors show that every matrix in \(SL_ n(A)\) is a product of two commutators, provided \(n\geq 3\) and A is commutative. The paper also contains a historical survey on commutators.
In preparation the authors show that any matrix in \(GL_ n(A)\) is similar to a product of a lower and an upper triangular matrix and also to a product of two matrices, each similar to a companion matrix.
Reviewer: E.Ellers

MSC:
15A23 Factorization of matrices
15B33 Matrices over special rings (quaternions, finite fields, etc.)
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