Corollary 2

Let A be a ring, and let $n$ be an integer $\geqslant 0$. The center of $\mathbf{M}_n(A)$ consists of the scalar matrices with entries in the center of A. We view $A^n$ as a left $\mathbf{M}_n(A)$-module (II, §10, No. 7, p. 349). The endomorphisms of this module are the mappings $x\mapsto xa$, where $a$ runs through A.

Let M be the right A-module $A^n_d$. It is balanced by Corollary 1. Consequently, the centers of $A_M, A'_M$, and $A''_M$ coincide. Corollary 2 then follows from the fact that $A_M$ is identified with A and $A'_M$ with $\mathbf{M}_n(A)$.