Let $R$ be a (not necessarily commutative) ring with $1$, and let $e$ be an idempotent, i.e. $e^2=e$. We call $e$ primitive if it is not the sum of two nonzero orthogonal idempotents.
Assume that $e$ is central in $R$. Is it then true that $e$ is primitive if and only if it is not the sum of two nonzero orthogonal central idempotents?
Of course, the "only if" part is obvious, but I wonder if the "if" part also holds. I tried to come up with counterexamples but could not find any.
No: for example in $M_2(\mathbb R)$, the only central idempotents are the identity and zero, so it is trivially not decomposable into two nonzero central orthogonal idempotents, but there are numerous ways to decompose it with noncentral orthogonal idempotents.
For example: $I_2=\begin{bmatrix}1&0\\0&0\end{bmatrix}+\begin{bmatrix}0&0\\0&1\end{bmatrix}$
For this reason, I believe authors do use the term "centrally primitive idempotent" to distinguish this from the usual "primitive idempotent."