I need some help showing that if S and T are subrings of R, then the union of S and T is a subring of R. Any ideas how to start?
This is a problem solved in a straightforward way by following the definitions. I start you off. If $\displaystyle a,b\in S\cap T$ then $\displaystyle a\in S, a\in T$ and $\displaystyle b\in S,b\in T$. Thus, $\displaystyle a+b\in S,a+b\in T$ because $\displaystyle S,T$ are subrings. And so $\displaystyle a+b\in S\cap T$. Thus, $\displaystyle S\cap T$ is closed under addition. Now you need to check the other conditions.