The existence of polynomials with given roots over non-commutative rings

This paper studies the problem of the existence of polynomials with given roots over associative non-commutative rings. It is shown that for arbitrary $n$ elements of an associative division ring there exists a polynomial of degree $n$ whose roots are these elements. The sufficient conditions for the existence of such a polynomial for elements of an arbitrary (not necessarily division) associative ring with unity are determined. For polynomials defined over a ring of square matrices over a field, a criterion for the existence of a second-degree polynomial with given roots is obtained, and examples of constructing polynomials with given roots are given.