A Universal Noncommutative Splitting Algebra for Polynomials
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.…
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