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.

In this note we show that all parabolic subgroups of Vershik-Kerov’s group $GLB(R)$ (i.e. subgroups containing $T(\infty ,R)$ —the group of infinite dimensional upper triangular matrices) are net subgroups for a wide class of semilocal rings $R$ .