Was it an even deal for me? Completely changing the question after posting it (without leaving any hint that the question has changed) is considered bad form.
Why are the centers of $R$ and $\text{End}_{\text{Ab}}(R)$ isomorphic? Well that's strange. I can't see any error in what you wrote; have I misunderstood the exercise? Here is the text exactly as it appears: "2.17. Let $R$ be a ring, and let $E = \text{End}_{\text{Ab}}(R)$ be the ring of endomorphisms of
the underlying abelian group $(R, +)$. Prove that the center of $E$ is isomorphic to the center of $R$."
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