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 (German: On the theory of associative number systems) and later rediscovered by Emmy Noether.
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 BxB = (x) 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 of finite dimension, 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[1][2] In particular, every automorphism of a central simple k-algebra is an inner automorphism.
[3][4] First suppose
End
{\displaystyle B=\operatorname {M} _{n}(k)=\operatorname {End} _{k}(k^{n})}
Then f and g define the actions of A on
denote the A-modules thus obtained.
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
By the first part applied to the maps
, we find for all z.
this time we find which is what was sought.