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 central variable. A pseudo-root polynomial p(t)=p0+p1t+?pktk an element ??R, for which there exist q1,q2 such that p=q1(t??)q2. We investigate the rational relationships hold among pseudo-roots p(t) by using diamond operations cover graphs modular lattices. When R division ring, each finite subset S corresponds to unique minimal monic fS vanishes on S. By results Leroy and Lam [16], set {fT:T?S} with right-divisibility order forms lattice join operation corresponding (left) least common multiple meet (right) greatest divisor. The edges graph this correspond naturally ?S fS. Given arbitrary ?S, our provide theoretic criterion guarantees rationally generates all particular,