I'm reading the proof of this theorem and there's one part that bothers me a bit. The theorem and the proof of that part (highlighted), are below.
Proof of Highlighted Part:
The definition, given in the text, of primitive central idempotent element $e$ is if $e$ is central and has no proper decomposition as a sum of orthogonal central idempotent elements.
The part that confuses me is I don't see how $e = e e_i + e(1-e_i)$ implies that $ee_i = e_i$ or $ee_i = 0$. It seems like the author was using the fact that $e$ is a primitive central idempotent to conclude this but in the statement we only assume $e$ is a central idempotent.
Also, assume that $e$ is a primitive central idempotent, then $e = e e_i + e(1-e_i)$ would imply that either $ee_i = 0$ or $e(1-e_i) = 0 $ but this would imply that $e = ee_i$ though.
Maybe I'm missing some trivial thing here but I don't see it.
Thank you all.
Best Answer
I'm not sure if the author has a typo or if I'm the one missing something, but we can write
$$e_i=ee_i+(1-e)e_i.$$
This is a sum of orthogonal central idempotents, and since $e_i$ is primitive, one of the terms is zero. Thus either $ee_i=0$ or $(1-e)e_i=0$, in which case $e_i=ee_i$.