Remark 2

Under the assumptions of Proposition 1, the A-module P is balanced; since the rings $A_P$ and $A'_P$ have the same center (VIII, p. 77), there is an isomorphism $\varphi$ from the center Z(A) of the ring A to the center Z(B) of the ring B, determined by the relation $\varphi (z)_P=z_P$ for $z\in Z(A)$. Moreover, the endomorphisms of the $(A,B)_k$-bimodule P are the homotheties $z_P$ where $z$ runs through Z(A); the automorphisms of the $(A,B)_k$-bimodule P are the homotheties $z_P$ where $z$ is invertible in Z(A).