Theorem 3 (Jacobson)

Let M be a semisimple A-module, and let $c$ be an endomorphism of the additive group M. Then $c$ belongs to the bicommutant $A''_M$ of M if and only if it satisfies the following condition:

(D) For every finite subset F of M, there exists an element $a$ of A such that $c$ coincides with $a_M$ on F.

First, suppose that $c$ satisfies condition (D). Let $u$ be an element of $A'_M$. Let $x$ be an element of M, and apply condition (D) to the subset $F =\{x, u(x)\}$. There exists an element $a$ of A such that $c(x) =ax$ and $c(u(x)) =au(x)$, so that $u(c(x)) =u(ax) =au(x) =c(u(x))$. Since $x$ is arbitrary, we have $cu=uc$; as this holds for all $u$, we have $c\in A''_M$.

For the converse, we use the following lemma.