Ideals of regular functions of a quaternionic variable

TL;DR

This paper proves that the ideal generated by multiple non-zero common slice regular functions over quaternions equals the entire ring, highlighting unique non-commutative algebraic properties.

Contribution

It establishes a quaternionic analogue of a classical ideal generation result, revealing new non-commutative syzygy structures for slice regular functions.

Findings

Ideal generated by non-zero slice regular functions is the whole ring

Non-commutative syzygies differ from complex case

Provides algebraic insights into quaternionic regular functions

Abstract

In this paper we prove that, for any $n\in \mathbb N$, the ideal generated by $n$ slice regular functions $f_1,\ldots,f_n$ having no common zeros concides with the entire ring of slice regular functions. The proof required the study of the non-commutative syzygies of a vector of regular functions, that manifest a different character when compared with their complex counterparts.