Lemma 4

Let V be a B-module and E the endomorphism $k$-algebra of V. Let $e$ be a projector of V and P the image of $e$. The mapping that sends $v\in eEe$ to the endomorphism $x\mapsto v(x)$ of the B-module P is an isomorphism of $k$-algebras from $eEe$ to End$_B(P)$.

Denote the mapping described in the statement by $\varphi :eEe\rightarrow$ End$_B(P)$; it is a homomorphism of $k$-algebras. Let $u\in$ End$_B(P)$. Denote by $v$ the endomorphism of V defined by $v(x) =u(e(x))$ for $x\in V$. We have $(eve)(x) =$ $u(x)$ for $x\in P$, that is, $\varphi (eve) =u$. Consequently, $\varphi$ is surjective. Let $w$ be an element of the kernel of $\varphi$; the restrictions of $w$ to the kernel and to the image of $e$ are zero, so $w$ is zero, which proves that $\varphi$ is injective.