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?

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- Feb 9th 2009, 06:03 PMEagle3Unions 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?

- Feb 9th 2009, 07:18 PMThePerfectHacker
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.

- Feb 9th 2009, 08:09 PMEagle3Union of subrings
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?

- Feb 9th 2009, 08:33 PMThePerfectHacker
- Feb 9th 2009, 08:35 PMEagle3Union of subrings
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