= Solution
The <Artin–Wedderburn theorem> says that a finite-dimensional <semisimple algebra> over $\mathbb C$ is a finite product $\prod_jM_{d_j}(\mathbb C)$: the division algebras in the general theorem reduce to $\mathbb C$ because the base field is algebraically closed. Each factor has <center of an associative algebra> $\mathbb C I_{d_j}$. Consequently a central element corresponds to $(\lambda_j I_{d_j})_j$. If it is <nilpotent>, then $\lambda_j^n=0$ for every $j$, whence every $\lambda_j=0$. Therefore $\boxed{z=0}$. The centrality assumption matters: a full <matrix algebra> has nonzero <nilpotent> elements.
Back to article page
