On real one-sided ideals in a free algebra

TL;DR

This paper investigates noncommutative real ideals in free algebras, identifying classes where the ideal equals its vanishing radical, and provides an algorithm to determine radicality.

Contribution

It extends classical radical notions to noncommutative free algebras, characterizes when ideals coincide with their radicals, and offers a computational tool.

Findings

Complete characterization for monomial and homogeneous principal ideals.

Analysis of principal univariate ideals with degree two generators.

Development of an algorithm to check radicality of ideals.

Abstract

In classical and real algebraic geometry there are several notions of the radical of an ideal I. There is the vanishing radical defined as the set of all real polynomials vanishing on the real zero set of I, and the real radical defined as the smallest real ideal containing I. By the real Nullstellensatz they coincide. This paper focuses on extensions of these to the free algebra R<x,x^*> of noncommutative real polynomials in x=(x_1,...,x_g) and x^*=(x_1^*,...,x_g^*). We work with a natural notion of the (noncommutative real) zero set V(I) of a left ideal I in the free algebra. The vanishing radical of I is the set of all noncommutative polynomials p which vanish on V(I). In this paper our quest is to find classes of left ideals I which coincide with their vanishing radical. We completely succeed for monomial ideals and homogeneous principal ideals. We also present the case of principal univariate ideals with a degree two generator and find that it is very messy. Also we give an algorithm (running under NCAlgebra) which checks if a left ideal is radical or is not, and illustrate how one uses our implementation of it.