Throughout this two-part note is a field and is a finite dimensional central simple -algebra. That means, as we mentioned in Remark 1 in this post, for some finite dimensional central -division algebra i.e. the center of is and We are going now to prove a result due to Skolem and Noether: every automorphism of is inner. That means if is a -algebra isomorphism, then there exists a unit such that for all Note that if is a simple -algebra and is any -algebra, then any nonzero -algebra homomorphism is injective. The reason is that is an ideal of and since is simple, we must either have or Since is nonzero, and so i.e. is injective. In particular, if is a finite dimensional simple -algebra, then

**Remark 1**. Let where is a division ring. By the Wedderburn-Artin theorem, has only one simple -module, say up to isomorphism. This is nothing but the set of all vectors with entries in It is also true that Since is semisimple, every -module is a finite direct product of Thus if is any -module, then for some integer

**Remark 2**. Let Since we may look at an element as both an matrix and a -linear transformations We will use both of them in the proof of the following lemma.

**Lemma**. Let and suppose that is any simple -subalgebra of If is a -algebra homomorphism, then there exists an invertible element such that for all

*Proof*. Clearly is Artinian because it is finite dimensional over Let be the unique simple -module (see Remark 1). Let Clearly is a left -module, and hence a left -module, where the multiplication of an element of by an element of is just the multiplication of an matrix by an vector. So by Remark 1,

for some integer We can also define the multiplication of an element of by an element of in this way:

With this definition will have a second structure as an -module because is a ring homomorphism. Let’s put So, by Remark 1,

for some integer Note that as -modules, and have the same structure because if then and hence by (2), Thus But by (1), and by (3), Hence and so, as -modules, Let be an -module isomorphism. Then obviously is also an isomorphism between -vector spaces and so is an invertible element of Finally, let and Let Then

So