Abstract
We investigate the commutators of elements of the group of infinite unitriangular matrices over an associative ring with and a commutative group of invertible elements. We prove that every unitriangular matrix of a specified form is a commutator of two other unitriangular matrices. As a direct consequence, we give a complete characterization of the lower central series of the group including the width of its terms with respect to basic commutators and Engel words. With an additional restriction on the ring , we show that the derived subgroup of coincides with the group . These results generalize the results obtained for triangular groups over a field.
Highlights
Let R be an associative ring with 1 and R∗ be its group of invertible elements
For a commutative ring R by SLV K (∞, R) we denote the subgroup of GLV K (∞, R) consisting of all matrices of the form (1), where G1 is a matrix from the special linear group SL(n, R) (n ∈ N), and G2 ∈ UT(∞, R)
We note that if R = K, where K is a field, one can prove a stronger result that every infinite unitriangular matrix A ∈ UT(∞, K) is a commutator of two infinite triangular matrices
Summary
Let R be an associative ring with 1 and R∗ be its group of invertible elements. By T(∞, R) (and T(n, R)) we denote the group of upper triangular matrices indexed by N × N (of size n × n, respectively), whose inverses are upper triangular. The above theorem has few important consequences, which we discuss in detail in the last part of our paper It has direct implications on the structure of the lower central series of groups T(∞, R) and UT(∞, R) and on the respective width of their terms. Our Basic Theorem allows for generalization of this result to the group of infinite matrices over certain rings. If R is an associative ring with 1, such that R∗ is commutative, the lower central series of the group UT(∞, R) is the sequence of subgroups: UT(∞, R) ≥ UT(∞, 1, R) ≥ UT(∞, 2, R) ≥ . We indicate some direct implications of Theorem 1 on the lower central series of the discussed groups of triangular matrices. By GL(∞, n, R) we denote the subgroup of GLV K (∞, R) consisting of all matrices of the form
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