Generators of ideal of coordinate axes in A^3

Question

So I know that the algebraic set $X$ equal to the union of the three coordinate axes in $\mathbb{A}^3$ is $I(X) = (xy, yz, xz)$ and that this is the fewest number of generators. But it seems that the ideal $(xyz, xy+yz+xz)$ has the same zero locus, but this supposedly is impossible by the embedding dimension being at least 3. What gives?

Answer 1

What gives is that $J = (xyz,xy+yz+xz)$ is not a radical ideal. In fact, $xy\notin J$, but $x^2 y^2 = xy(xy+yz+xz)-(x+y)xyz \in J$. So we can't use these equations to cut out $X$ without some multiplicity.