When a given ideal is a radical ideal

Question

I am wondering if there are any canonical methods for checking whether a given ideal is radical. For example, I got stuck on the following example:

Let $f=x+2y-z$ and $g=z-2w$ and let $I$ and $J$ be the ideals in $\Bbb{C}[x,y,z,w]$ generated by $f$ and $g$ respectively. Is the product ideal $K=I\cdot J$ radical ?

Answer 1

Suppose that $a^k \in (fg)$, so $a^k \in (f)$ and $a^k \in (g)$. Since $(f)$ and $(g)$ are prime ideals, we have $a\in (f)\cap (g) = (fg)$.

In general, showing that an ideal is radical is not an easy problem. One should expect to approach a general problem using Gröbner bases.