A ring $R$ is called abelian if all idempotents in $R$ are central. So, every ring should have a universal abelian quotient ring.
One way to do this is to construct an ascending chain of ideals $I_0 \subseteq I_1 \subseteq I_2 \subseteq ...$ where $I_0=0$ and $I_{n+1}$ is the two-sided ideal of $R$ generated by all commutators $ex-xe$ for $e,x \in R$ with $e^2-e \in I_n$. Then, the universal abelian quotient ring of $R$ should be the quotient of $R$ by the union of the ideals $I_n$ for $n \ge 0$.
But is $I_2$ always equal to $I_1$? That is, is centralizing all idempotents just once enough to make a ring abelian?
Perhaps, one could have $e^2-e=fx-xf$ where $f^2=f$ but $e+I_1$ does not "lift" to an idempotent in $R$. If this is the case, then centralizing all idempotents in $R$ could lead to new idempotents that would then also need to be centralized.