See also The double centralizer theorem.

Throughout this post is a field and as usual, is the algebra of matrices with entries in

**Notation.** Let be a ring and let be a subring of Let Then we will denote by the centralizer of in i.e. If then we will write instead of

The following lemma is a well-known result in linear algebra and so I will not prove it here.

**Lemma.** Let and let If then In other words, for all

The lemma can be extended to any finite dimensional central simple algebra:

**Theorem.** (W. L. Werner, 1969) Let be a finite dimensional central simple -algebra and let If then In other words, for all

*Proof***.** Let We proved in here that for some integer

*Claim *.

*Proof of the claim*. Let be a -basis for and let So

for some Since we have

So and hence, since are -linearly independent, we have for all That means for all Therefore

Thus

But clearly if then

So and the claim now follows from

So, by the lemma and the claim, Thus and the proof of the theorem is complete.

**Remark**. Note that where means the center of A consequence of the theorem is that for any , is commutative iff