math.RA — Rings and Algebras

A Universal Noncommutative Splitting Algebra for Polynomials

Contributed by Darij Grinberg

Let $R$ be a commutative ring. We show that every homogeneous polynomial $f\in R[x_{1},\ldots,x_{m}]$ of degree $n$ splits into a product of $n$ homogeneous linear forms over a suitable noncommutative ring extension $S$ of $R$ (that is, over a noncommutative $R$-algebra $S$ whose structure morphism $R\rightarrow S$ is injective). The algebra $S$ is universal for such factorizations and is free as an $R$-module. The proof uses Bergman's Diamond Lemma: we define $S$ by generators and relations, and the relations form a terminating reduction system that has no ambiguities and thus is confluent. An analogous result is also shown for inhomogeneous polynomials (with inhomogeneous factors). This easily follows from the homogeneous case by homogenizing and then setting the homogenizing variable equal to $1$.