A criterion for the universality of a matrix from the group $UT_n(R)$ over a commutative and associative ring $R$ with unity

We find necessary and sufficient universality conditions of a matrix from the unitriangular matrix group of arbitrary finite dimension over a commutative associative ring with unity. An algorithm is used to determine the universality of the element of the unitriangular matrix group over the ring of polynomials with a finite number of variables with integer coefficients.