Proposition 12

Let B be a simple ring and A a simple subring of B. Suppose that B is a finitely generated left A-module. Let M be a nonzero finitely generated left B-module. Set $A'=$ End$_A(M)$ and $B'=$ End$_B(M)$. Then $B'$ is a subring of $A'$, the rings $A'$ and $B'$ are simple, $A'$ is a finitely generated left $B'$-module, and we have the equalities

By Proposition 4 of VIII, p. 123, the ring $A'$ is simple, M is an $A'$-module of finite length, and we have

long$_A(M) =$ long(A$'$), long$_{A'}(M) =$ long(A).

For the same reasons, the ring $B'$ is simple, and we have

long$_B(M) =$ long(B$'$), long$_{B'}(M) =$ long(B).

By formulas (31) and (30), we therefore have

$h(B,A) =$ long$_A(M)/$ long$_B(M) =$ long(A$'$)$/$ long(B$'$) $=i(A',B')$;

the equality $h(A',B') =i(B,A)$ can be established analogously. From this, we deduce

using formula (32). In particular, $A'$ is a finitely generated left $B'$-module.