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It is certainly possible to concoct an artificial example where $x\mapsto rx$ is a homomorphism but $r$ does not commute with everything.

The elements shall be non-commutative polynomials in two variables, say $x$ and $y$, such that no term of the polynomial contains more than one occurrence of $x$. Addition is ordinary polynomial addition. For multiplication, multiply the polynomials as usual and then delete from each term the second occurrence of $x$, if there is one.

For example:

$\begin{array}{rcl}(2y^2xy + x + 2y)(xy+yx)&=&2y^2xyxy + x^2y + 2yxy + 2y^2xy^2x + xyx + 2y^2x \\&=& 2y^2xy^2 + xy + 2yxy + 2y^2xy^2 + xy + 2y^2x \\&=& 4y^2xy^2 + 2xy + 2yxy + 2y^2x\end{array}$

In other words, this is the free ring on $x$ and $y$ quotiented by the relations $xab=xaxb$ for all $a$ and $b$.

Clearly $x$ does not commute with $y$, so is not in the center, and yet $xab=(xa)(xb)$ for all $a$ and $b$.