# Centers of Rings

• December 3rd 2009, 04:48 PM
smacktalk88
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?
• December 3rd 2009, 05:30 PM
aliceinwonderland
Quote:

Originally Posted by smacktalk88
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.