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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- Dec 3rd 2009, 04:48 PMsmacktalk88Centers 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?

- Dec 3rd 2009, 05:30 PMaliceinwonderland
The center of a ring A, written C(A), is $\displaystyle \{a \in A | ax=xa \text{ for all } x \in A\}$.

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