Example

$*$ Let A be a $\mathbf{C}$-algebra generated by a countable family of elements; it has countable dimension over $\mathbf{C}$. Let M be a simple A-module; it is monogenous, so admits a countable basis over $\mathbf{C}$. Since the field $\mathbf{C}$ is not countable (Gen. Top., IV, §4, No. 1, p. 44), we have $[M :\mathbf{C}]<$ Card($\mathbf{C}$). Therefore the endomorphisms of the A-module M are the homotheties $\alpha_M$ with $\alpha \in \mathbf{C}$. This applies, in particular, when A is the universal enveloping algebra of a finite-dimensional Lie algebra over $\mathbf{C}($Lie, I, §2, No. 7, p. 21, Corollary $3).*$