Let $k$ be a field of characteristic zero, $R$ a commutative $k$-algebra, and $E$ the monoid of $k$-endomorphisms of $R$. Assume that $e_i \in E$, $1 \leq i \leq 2$, has centralizer $C_{e_i}$ in $E$ ($C_{e_i}$ is of course a sub-monoid of $E$).
Is it possible to know what is the centralizer of $e:=e_1e_2$? Clearly, $C_{e_1} \cap C_{e_2}$ is contained in $C_{e}$, the centralizer of $e$ (but $C_{e}$ may strictly contain $C_{e_1} \cap C_{e_2}$).
(What if we further assume that $e_1e_2=e_2e_1$?).
Any relevant ideas/hints are welcome. Perhaps the first answer in this question is relevant to my question.
I doubt there is very much that is interesting you can say about this without adding some strong additional hypotheses. As an illustrative example, suppose $e_1$ is an automorphism of $R$ and $e_2=e_1^{-1}$. Then $e_1e_2=e_2e_1$, but $C_{e_1e_2}$ consists of all of $E$ since $e_1e_2=1$. On the other hand, $C_{e_1}\cap C_{e_2}=C_{e_1}$ will typically be smaller than $E$.