Skolem–Noether theorem
In mathematics, the Skolem–Noether theorem, named after Thoralf Skolem and Emmy Noether, is an important result in ring theory which characterizes the automorphisms of simple rings.
The theorem was first published by Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme (German: On the theory of associative number systems) and later rediscovered by Noether.
Skolem-Noether theorem
In a general formulation, let A and B be simple rings, and K = Z(B) be the centre of B. Suppose that the dimension of B over the field K is finite, that is B is a central simple algebra (K is a field since for any nonzero x in K, Bx is a nonzero, two-sided ideal of B, so simplicity of B implies that Bx = B and hence x is invertible).
Then if
- f,g : A → B
are K-algebra homomorphisms, there exists a unit b in B such that
- g(a) = b·f(a)b−1
for all a in A.
Implications
- Every automorphism of a Brauer algebra is an inner automorphism.
References
- Thoralf Skolem, Zur Theorie der assoziativen Zahlensysteme, 1927
- A proof [1]