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 then and . Thus, because are subrings. And so . Thus, is closed under addition. Now you need to check the other conditions.
Are you saying that the same argument holds for S U T. If so, isn't there a possibility that a+b could be in R, but not in S U T?
Intersection of subrings is always a subring (look at above post). However, the union of subrings is not always a subring unless one ring is contained in another.