A note on irreducible slice algebraic sets

TL;DR

This paper proves that for certain ideals in quaternionic slice regular polynomials, the associated algebraic sets are irreducible, establishing a link between radical ideals and irreducibility.

Contribution

It establishes a characterization of irreducible algebraic sets in quaternionic slice regular polynomials via quasi prime ideals.

Findings

Symmetrization of common zero sets is irreducible for right radical quasi prime ideals.

For radical ideals, the zero set is irreducible if and only if the ideal is quasi prime.

The result connects algebraic properties of ideals with geometric irreducibility in quaternionic polynomial rings.

Abstract

In this short note we prove that if $I$ is a right radical and quasi prime ideal in the ring of quaternionic slice regular polynomials, then the symmetrization $\mathbb S_{V_c(I)}$ is an irreducible algebraic set, where $V_c(I)$ is the set of common zeros with commuting components of polynomials in $I$. Combining this fact with the results proved in our previous paper [3], we obtain that for $I$ radical, $V_c(I)$ is irreducible if and only if $I$ is quasi prime.