Definition 6

Two square matrices X, X' of order n over a ring A are called similar if there exists an invertible square matrix P of order n such that

Clearly the relation "X and X' are similar" is an equivalence relation on $M_n(A)$ meaning that X and X' are transformed into one another by an inner automorphism of this ring.

With this definition, Corollary 1 to Proposition 6 of no. 8 can be stated by saying that when the basis of an A-module E (with a finite basis) is changed, the matrix of an endomorphism u with respect to the new basis is similar to the matrix of u with respect to the old basis.