In Martin Isaacs' Character Theory of Finite Groups, the author states on p. 3 that:
Let $V$ be an $A$-module [$A$ being a $\mathbb{F}$-algebra]. Each $x \in A$ defines a map $x_V : V \rightarrow V$ by $v \mapsto vx$. [...] The map $x \mapsto x_V$ is an algebra homomorphism $A \rightarrow \text{End}(V)$. Its image is denoted $A_V$.
Later on, on the same page, he states:
If $V$ and $W$ are $A$-modules, a linear transformation $\varphi:V \rightarrow W$ is an $A$-homomorphism if $\varphi(vx) = \varphi(v)x$ for all $v \in V$ and $x \in A$. [...] The set $\text{Hom}_A (V,W)$ of $A$-homomorphisms from $V$ to $W$ has the structure of an $\mathbb{F}$-space by $(c \varphi) (v) = c(\varphi (v))$ for $c \in F$ and $(\varphi + \vartheta) (v) = \varphi (v) + \vartheta (v)$. In addition, $\text{Hom}_A (V,V)$ is a ring [remember, $\varphi \theta$ is defined by [composition of functions]], and in fact $\text{Hom}_A (V,V)$ is an $\mathbb{F}$-algebra. It is exactly the centralizer of $A_V$ in $\text{End} (V)$ and is denoted $\textbf{E}_A (V)$.
However, Isaacs never explicitly proves that $\text{Hom}_A (V,V)$ is exactly the centralizer of $A_V$ in $\text{End} (V)$.
I'd very much appreciate it if someone with more experience in this topic than me could please explain how one goes about proving that.