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 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.Comment: v1: 31 pages; v2: 32 page