Remark 1

Suppose that A is a simple subring of a simple ring B and that the right degree $[B : A]_d$ is finite. Let C be the endomorphism ring of B viewed as a right A-module; it is a simple ring by Proposition 4, b) of VIII, p. 123. For any $b$ in B, let $\gamma (b)$ be the mapping $x\mapsto bx$ from B to B; then $\gamma :b\mapsto \gamma (b)$ is an isomorphism from B to a subring of C. Moreover, if $(x_1, . . . , x_m)$ is a basis of the right A-module B, then the morphism that sends $c$ to $(c(x_1), . . . , c(x_m))$ is an isomorphism of left B-modules from C to $B^m_s$; consequently, we have the relation

Taking into account relation (2) of VIII, p. 123 applied to the right A-module B, we see that

(5) long(C) $= [B : A]_d$ long(A).