Definition 2

Let B be a ring and A a simple subring of B. The dimension of the free left A-module B is called the (left) degree of B over A and is denoted by[^1] $[B : A]_s$.

By replacing A and B with the opposite rings, we deduce from the above that B is a free right A-module. We denote its dimension by $[B : A]_d$ and call it the right degree of B over A.

We can give an example of a field B and a subfield A such that the degrees $[B : A]_s$ and $[B : A]_d$ differ.[^2]

Let B be a ring, A a simple subring of B, and S a simple left A-module. Let M be a left A-module and $\mathfrak{a}$ its length. The A-modules M and $S^{(\mathfrak{a})}$ are isomorphic (VIII, p. 122, Proposition 2), hence so are the B-modules $B\otimes_AM$ and $(B\otimes_AS)^{(\mathfrak{a})}$. The relation

(3) long$_A(B\otimes_AM) = [B : A]_s$ long$_A(M)$

follows from Proposition 5 and Definition 2.