Proposition 6

Let C be a ring, B a simple subring of C, and A a simple subring of B. We then have $[C : A]_s= [C : B]_s[B : A]_s$.

Let us introduce a basis $(e_i)_{i\in I}$ of C viewed as a left B-module and a basis $(f_j)_{j\in J}$ of B viewed as a left A-module. Then the family $(f_je_i)_{j\in J,i\in I}$ is a basis of C viewed as a left A-module (II, §1, No. 13, p. 222, Proposition 25); Proposition 6 follows.