exercise in Shafarevich

Question

I didn't know what is the meaning of two elements, I just found that xy+xz+yz=0 and xy+xz-yz=0 can satisfy the requirement.

Shafarevich--Basic Algebraic Geometry 1 P80

Answer 1

I'm guessing you're claiming that $xy+xz+yz$ and $xy+xz-yz$ generate the ideal because any zero of both equations lies on one of the three axes, and that those two equations vanish on the three axes. By the nullstellensatz, that means that the radical of the ideal generated by those two elements is the ideal of the curve whose components are the $3$ coordinate axes.

However, this ideal is not radical. Notice that $xy \notin I$ but $(xy)^2 \in I$.