Radicals and Ideals

Question

Can anyone help me to solve this problem and explain how to find the right solution.

Let $A=K[x,y,z]$. Find the radical of $I=(xyz, xy+xz+yz, x+y+z)$.

Thanks.

Answer 1

Just use that the radical of $I$ is the intersection of the prime ideals in $k[x,y,z]$ which contain $I$, so you first have to determine which are those prime ideals.

Now if such an ideal $\mathfrak p$ contains $I$, it contains $xyz$, hence one of the factors – say it contains $x$. As a consequence, it also contains $(xy+yz+zx)-x(y+z)=yz$, hence $y$ or $z$. If it contains $y$, it also contains $(x+y+z)-x-y=z$, and similarly if it contains $z$, it necessarily also contains $y$. To sum all this up, $\mathfrak p$ contains $x, y$ and $z$.

As the generators of $I$ are symmetric in $x, y,z$, it will be the same argument if $\mathfrak p$ contains the factor $y$ or $z$. Thus the only prime ideal which contains $I$ is the maximal ideal $\langle x, y,z\rangle$, which is therefore the radical of $I$.