Universal elements of unitriangular matrices groups

Abstract

The following theorems are proved for a matrix g from the group of unitriangular matrices over a commuta- tive and associative ring K of finite dimension of greater than three with unity: 1) if the matrix g is universal then all of its elements are on the first collateral diagonal except extreme ones are nonzero; 2) if all elements of the first collateral diagonal of the matrix g, with the possible exception of the last element are reversible in K, then g is universal; 3) if the ring K is Euclidean and has no reversible elements except trivial ones, then it follows from the universality of the matrix g that all the elements of its first collateral diagonal, except the extreme ones, are reversible in K.