  In this paper, we study the endomorphism properties of vertex operator
algebras over an arbitrary field $\mathbb{F}$, with $\text{Char}(\mathbb{F})
\neq 2$. Let $V$ be a strongly finitely generated vertex operator algebra over
$\mathbb{F}$, and $M$ be an irreducible admissible $V$-module. We prove that
every element in $\text{End}_V(M)$ is algebraic over $\mathbb{F}$ and that
$\text{End}_V(M)$ is also finite-dimensional. As an application, we prove
Schur's lemma for strongly finitely generated vertex operator algebras over
arbitrary algebraically closed fields, and we give a test for absolute
irreducibility of $V$-modules.
