Prove or disprove: if $S$ is a subring of $R$ then $Z(S)$ is a subring of $Z(R)$

Question

Prove or disprove: if $S$ is a subring of $R$ then $Z(S)$ is a subring of $Z(R)$

I think this will be false because if something commutes with all elements of $S$ then it does not necessarily commute with all elements of $R$. But I'm really struggling with finding a counterexample.. any hints?

Answer 1

Hint:

The complex numbers $\mathbb C$ form a subring of Hamilton's quaternions $\mathbb H$.