Radical ideal in $\mathbb{R}[x,y,z]$

Question

In $\mathbb{R}[x,y,z]$ is the ideal $I=\left\langle xz,yz\right\rangle$ radical?

If $f \in I$ tried write $f=g.xz+h.yz+ax+by+c$ and conclude that $f^m \notin I$, if $m>0$, but I could not.

Answer 1

Let's note $I$ is the intersection of the primes $\langle x,y \rangle$ and $\langle z\rangle$ of $\mathbb{R}[x,y,z]$.

Now is easy to conclude using the following facts:

• A prime ideal $P$ is radical (The quotient is a domain and so is reduced)
• The radical of finite intersection of ideals is the intersection of the radicals (A containment is easy, for the other one check it by taking two elements of the intersection of radicals)