Remark 3

Let A be a simple ring, S a simple A-module, and D the opposite ring of the field End$_A(S)$. Then the A-module $A_s$ has finite length, and dim$_D(S)$ is equal to long(A). Indeed, by Lemma 1, the ring A is isomorphic to End$_D(S)$; we then apply Lemma 2.