Zariski closure of the boundary of the first quadrant

Question

What is the Zariski closure of the boundary of the first quadrant in $\mathbb{R}^2$?

I know that the boundary is $S=\{(a,0)\in\mathbb{R}^2:a\geq0\}\cup\{(0,b)\in\mathbb{R}^2:b\geq0\}$. As per the definition of the Zariski closure, I suppose I need to find $\textbf{I}(S)$ and then find its variety. Is this the way to proceed? If so, what to do exactly to get the ideal and then the variety?

Answer 1

Take the ideal $I=\langle xy\rangle$ where $xy=0$ iff $x=0$ or $y=0$.