I am reading Serre's proof on the Galois cohomological interpretation of the Brauer group, i.e. on the isomorphism $ \operatorname{Br}(K/k) \to H^2(K/k)$ (Serre, Local Fields, X, § 5).
In Proposition 9, he claims it is already shown that the defined map $A(K/k) \to H^2(K/k)$ is injective.
This is clear to me for $A(n, K/k) \to H^2(K/k)$ from the long exact cohomology sequence. However I do not see why, taking the union over all $n$ yields an injective map too.
Seems to be a very simple argument I am missing, I appreciate your help.
Edit: this boils down to a statement like the following:
If C and D are central simple k-algebras, not necessarily of the same dimension, with $C \otimes D^{op}$ isomorphic to a matrix algebra over k, then $C \cong D$.
Is this statement correct?
This is the uniqueness of the inverse element in the classical definition of the Brauer group (via equivalence classes of central simple algebras).
See for example: Kersten, Ina: Brauergruppen von Körpern as a German language reference.