## Abstract

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)=p_{0}+p_{1}t+⋯p_{k}t^{k} is an element ξ∈R, for which there exist polynomials q_{1},q_{2} such that p=q_{1}(t−ξ)q_{2}. 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.

Original language | English (US) |
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Article number | 106581 |

Journal | Journal of Pure and Applied Algebra |

Volume | 225 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2021 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory

## Keywords

- Factorizations
- Lattices
- Non-commutative polynomials
- Roots