I want to find the radical of the ideal generated by $(2,2^2,...,2^n,...)$ in $\mathbb{Z^{\mathbb{N}}}$. Also want to know if the radical ideal is finitely generated or not.
My approach is to use the fact that the radical of an ideal $I$ is the intersection of all prime ideals containing $I$ but problem is that in this case I don,t know the complete set of prime ideals of $\mathbb{Z^{\mathbb{N}}}$. I know the list of prime ideals in $\mathbb{Z^{\mathbb{N}}}$ but I am not sure this list is complete or not namely, $P_i=\mathbb{Z}\times \dots \times p\mathbb{Z}\times\mathbb{Z}\times\dots$, where $p\mathbb{Z}$ is at ith component.
Thank you.