Skolem–Noether theorem

In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the automorphisms of simple rings. It is a fundamental result in the theory of central simple algebras.
The theorem was first published by Thoralf Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme and later rediscovered by Emmy Noether.

Statement

In a general formulation, let A and B be simple unitary rings, and let k be the center of B. The center k is a field since given x nonzero in k, the simplicity of B implies that the nonzero two-sided ideal is the whole of B, and hence that x is a unit. If the dimension of B over k is finite, i.e. if B is a central simple algebra, and A is also a k-algebra, then given k-algebra homomorphisms
there exists a unit b in B such that for all a in A
In particular, every automorphism of a central simple k-algebra is an inner automorphism.

Proof

First suppose. Then f and g define the actions of A on ; let denote the A-modules thus obtained. Since the map f is injective by simplicity of A, so A is also finite-dimensional. Hence two simple A-modules are isomorphic and are finite direct sums of simple A-modules. Since they have the same dimension, it follows that there is an isomorphism of A-modules. But such b must be an element of. For the general case, is a matrix algebra and that is simple. By the first part applied to the maps, there exists such that
for all and. Taking, we find
for all z. That is to say, b is in and so we can write. Taking this time we find
which is what was sought.