On the rational relationships among pseudo-roots of a non-commutative polynomial

For a non-commutative ring R, we consider factorizations of polynomials in R[t] where t is a central variable. A pseudo-root of a polynomial p(t)=p₀+p₁t+⋯p_kt^k is an element ξ∈R, for which there exist polynomials q₁,q₂ such that p=q₁(t−ξ)q₂. We investigate the rational relationships that hold among the pseudo-roots of p(t) by using the diamond operations for cover graphs of modular lattices. When R is a division ring, each finite subset S of R corresponds to a unique minimal monic polynomial f_S that vanishes on S. By results of Leroy and Lam [16], the set of polynomials {f_T:T⊆S} with the right-divisibility order forms a lattice with join operation corresponding to (left) least common multiple and meet operation corresponding to (right) greatest common divisor. The set of edges of the cover graph of this lattice correspond naturally to a set Λ_S of pseudo-roots of f_S. Given an arbitrary subset of Λ_S, our results provide a graph theoretic criterion that guarantees that the subset rationally generates all of Λ_S, and in particular, rationally generates S.