Thread: Unions of subrings

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?

Originally Posted by Eagle3

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?

Originally Posted by Eagle3

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.

That is what I thought. I just could find a way to prove that it was true and logically I could see where it might not be true.

Thanks