I need to find two subrings $S$ and $T$ of the polynomial ring $\mathbb{R}[x,y]$ such that $S+T$ is not a subring of $\mathbb{R}[x,y]$.
This is the ring of polynomials with real coefficients in the two variables $x$ and $y$. In order for $S+T$ to fail to be a subring, I need to show that it either doesn't contain $0$ or it isn't closed under negation, addition, or multiplication. But I'm having trouble coming up with subrings that will work.
Even hints would be helpful provided you're willing to answer follow-up questions!