The center of a ring R is { }
Let --> be a surjective homomorphism of rings. Prove that the image of the center of is contained in the center of .
Let be in the center of R ( ) . Then for any we can find a preimage in since is surjective. Then consider
.
Thus every element of the image of the center of R commutes with any element of S, so it must be contained in the center of S by definition.