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Math Help - subring

  1. #1
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    Brasilia
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    subring

    Hi ! i have this exercise , i need help



    For a fixed element a  \in{} \mathbb{R} we have  C(a)=r \in{} R\ /\ ra=ar . Prove that  C(a) is a subring of R containing a . Prove that the center of R is the intersection of the subrings  C (a) for all s \in{} R
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  2. #2
    MHF Contributor

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    Tejas
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    Re: subring

    in this case, i believe that you want to show this for an arbitrary ring (possibly non-commutative) R, *not* the ring of real numbers (because every real number centralizes every other real number, because multiplication is commutative on the reals).

    you need to show 3 things:

    1) if x,y are in C(a), then x - y is in C(a).
    2) if x,y are in C(a) , so is xy.
    3) C(a) is non-empty (so we can actually use (1) & (2)).

    proving a is in C(a) will show (3). does aa = aa?

    the distributive law should figure heavily in your proof of (1). the associative law of multiplication should figure heavily in your proof of (2).

    to prove that:

    C(R) = \bigcap_{a \in R} C(a),

    is equivalent to saying that for r in C(R), ra = ar for EVERY a in R. is that not the definition of C(R)?
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