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Math Help - Subring of R generated by X

  1. #1
    Super Member Bernhard's Avatar
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    Subring of R generated by X

    Given any ring R AND a subset X of R, we can consider the set of all subrings of R containing X and form their intersection, T, say.

    Show that T is the smallest subring of R containing X.
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  2. #2
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    Re: Subring of R generated by X

    Quote Originally Posted by Bernhard View Post
    Given any ring R AND a subset X of R, we can consider the set of all subrings of R containing X and form their intersection, T, say.

    Show that T is the smallest subring of R containing X.
    This is actually a straightforward manipulation of definitions. You'll need to prove two things: T is a subring, then T is the smallest subring containing X. Suppose your set of all subrings of R containing X is given by \mathcal{S}=\{R_\alpha |\alpha \in \Lambda\}.


    For the first one, you really just need to check that r,s\in T=\displaystyle{\bigcap_{\alpha\in \Lambda} R_{\alpha}}\implies r-s,rs\in T=\displaystyle{\bigcap_{\alpha\in \Lambda} R_{\alpha}}. The notation is a bit cumbersome, but this fact is actually quite trivial if you think about it. We also have X\subseteq T by definition of T.



    The second bit is even easier: Let R_\beta be a subring of R containing X. Since this is a subring of R containing X, we have R_\beta \in \mathcal{S} by definition of \mathcal{S}. Using the basic property A\cap B \Rightarrow A\cap B \subseteq A of intersections, this says that T\subseteq R_\beta
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