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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