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Math Help - Centers of Rings

  1. #1
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    Centers of Rings

    If I have an onto ring homomorphism from A to B, how do I know that the center of A is contained by the center of B?
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  2. #2
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    Quote Originally Posted by smacktalk88 View Post
    If I have an onto ring homomorphism f from A to B, how do I know that the center of A (f(A)?) is contained by the center of B?
    The center of a ring A, written C(A), is \{a \in A | ax=xa \text{ for all } x \in A\}.

    Since f is a ring homomorphism, we have f(ax)=f(xa) and f(a)f(x)=f(x)f(a), where a \in C(A) and x \in A. Since f is onto, f(a) should be contained in the center of B.
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