On real one-sided ideals in a free algebra

Abstract

In real algebraic geometry there are several notions of the radical of an ideal I. There is the vanishing radical I defined as the set of all real polynomials vanishing on the real zero set of I, and the real radical Ire defined as the smallest real ideal containing I. (Neither of them is to be confused with the usual radical from commutative algebra.) By the real Nullstellensatz, I=Ire. This paper focuses on extensions of these to the free algebra R〈x,x∗〉 of noncommutative real polynomials in x=(x1,…,xg) and x∗=(x1∗,…,xg∗).We work with a natural notion of the (noncommutative real) zero set V(I) of a left ideal I in R〈x,x∗〉. The vanishing radical I of I is the set of all p∈R〈x,x∗〉 which vanish on V(I). The earlier paper (Cimprič et al. [6]) [6] gives an appropriate notion of Ire and proves I=Ire when I is a finitely generated left ideal, a free ∗-Nullstellensatz. However, this does not tell us for a particular ideal I whether or not I=Ire, and that is the topic of this paper. We give a complete solution 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. We discuss an algorithm to determine if I=Ire (implemented under NCAlgebra) with finite run times and provable effectiveness.