Let $A$ be a finite dimensional $\mathbb C$-algebra. Let $e_1,\ldots,e_r\in A$ be nonzero idempotents (with $r>0$), i.e. $e_i^2=e_i$. My question is: Can it happen that $e_1+\cdots+e_r=0$? I can't think of a single example.
Note: I do not require the $e_i$ to be central, primitive, or orthogonal.
We can WLOG assume that the algebra $A$ is embedded in $\mathrm{M}_n\left(\mathbb C\right)$ for some $n\in\mathbb N$ (because the $A$-module $A$ is faithful and finite-dimensional, so that $A$ is embedded in $\mathrm{End}_{\mathbb C} A \cong \mathrm{M}_n\left(\mathbb C\right)$ for $n=\dim_{\mathbb C}A$). Then, $e_1, e_2, \dots, e_r$ are idempotent matrices, and have idempotent sum (because $0$ is idempotent). According to MathOverflow question #115067, any finite list of idempotent matrices over $\mathbb C$ (or any other field of characteristic $0$) having idempotent sum must be a list of orthogonal idempotents. Hence, your idempotents $e_1, e_2, \dots, e_r$ are orthogonal. Thus, $e_1\left(e_1+e_2+\cdots+e_r\right) = e_1e_1 + e_1e_2 + \cdots + e_1e_r = e_1 + 0 + \cdots + 0 = e_1$. Since $e_1+e_2+\cdots+e_r=0$, this rewrites as $e_1\cdot 0 = e_1$, so that $e_1 = 0$. This contradicts the assumption that $e_1, e_2, \dots, e_r$ are nonzero idempotents.