Homomorphic images of ring centers are central

Let \varphi : R \rightarrow S be a surjective ring homomorphism. Prove that \varphi[Z(R)] \subseteq Z(S).


Suppose r \in \varphi[Z(R)]. Then r = \varphi(z) for some z \in Z(R). Now let x \in S. Since \varphi is surjective, we have x = \varphi y for some y \in R. Now xr = \varphi(y)\varphi(z) = \varphi(yz) = \varphi(zy) = \varphi(z)\varphi(y) = rx. Thus r \in Z(S).

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