Unions of subrings

• Feb 9th 2009, 06:03 PM
Eagle3
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?
• Feb 9th 2009, 07:18 PM
ThePerfectHacker
Quote:

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 $\displaystyle a,b\in S\cap T$ 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 PM
Eagle3
Union 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 PM
ThePerfectHacker
Quote:

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.
• Feb 9th 2009, 08:35 PM
Eagle3
Union 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