On the irreducibility of slice algebraic sets

TL;DR

This paper explores the algebraic structure of slice algebraic sets in quaternionic space, establishing conditions for irreducibility and linking these geometric objects to algebraic ideals, with a focus on quasi prime ideals and their properties.

Contribution

It introduces algebraic criteria for irreducibility of slice algebraic sets and demonstrates the equivalence between irreducibility and quasi prime ideals in certain cases.

Findings

Radical ideals of irreducible sets are quasi prime.

Conditions on right ideals guarantee irreducibility.

Equivalence between irreducibility and quasi prime ideals for principal right ideals.

Abstract

In the present paper we investigate the relations between irreducible slice algebraic sets in $\mathbb{H}^n$ and quasi prime right ideals of the ring of slice regular polynomials in $n$ quaternionic variables. We provide algebraic conditions on right ideals of slice regular polynomials which guarantee the irreducibility of the corresponding slice algebraic sets and show that radical ideals associated with irreducible slice algebraic sets are quasi prime. Furthermore we establish that this correspondence is an equivalence in the case of principal right ideals.