Example of subring in $\mathbb Z_3\times \mathbb Z_3$ that's not an ideal

Question

Let $R$ be the ring $\mathbb Z_3\times \mathbb Z_3$ and $S=\{(n,n): n \in\mathbb Z_3\}.$

I'm skipping the step of proving $S$ is a subring of $R$ for now.

Then, let $ x=(1,1)$ and $r=(2,0)$, then, $x*r=(2,0) $ which is not in $S$. Then we know that $S$ is a subring but not an ideal. Is this example concrete? Thank you for the help