A Universal Noncommutative Splitting Algebra for Polynomials

Darij Grinberg

A Universal Noncommutative Splitting Algebra for Polynomials

A mix of human-written and AI-generated textHuman understanding: all partsmath.RA — Rings and Algebras math.AC — Commutative Algebra math.CO — Combinatorics

Contributed by Darij Grinberg ↗

Version 1 / Sep 29, 2026 / CC0 1.0

Abstract

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

.

Provenance statement

Written by GPT-5.5; proofread and edited by myself. This is also available on my website: https://www.cip.ifi.lmu.de/~grinberg/algebra/ncsplit-gpt.pdf & https://www.cip.ifi.lmu.de/~grinberg/algebra/ncsplit-gpt.tex .

Tools used

OpenAI: ChatGPTVersion 5.5

References

No bibliography entries.

Version history

v1Submitted by Darij GrinbergInitial depositCurrentSep 29, 2026