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Math Help - Unions of subrings

  1. #1
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    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?
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  2. #2
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    Quote Originally Posted by Eagle3 View Post
    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 a,b\in S\cap T then a\in S, a\in T and b\in S,b\in T. Thus, a+b\in S,a+b\in T because S,T are subrings. And so a+b\in S\cap T. Thus, S\cap T is closed under addition. Now you need to check the other conditions.
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  3. #3
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    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?
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  4. #4
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    Quote Originally Posted by Eagle3 View Post
    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.
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  5. #5
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    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.

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