Thread: 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?

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 .

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