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 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
f, g : A → B,
there exists a unit b in B such that for all a in A[1] [2]
g(a) = b · f(a) · b−1.
In particular, every automorphism of a central simple k-algebra is an inner automorphism.[3] [4]
First suppose
B=\operatorname{M}n(k)=
n) | |
\operatorname{End} | |
k(k |
kn
Vf,Vg
f(1)=1 ≠ 0
Vf,Vg
b:Vg\toVf
\operatorname{M}n(k)=B
B ⊗ kBop
A ⊗ kBop
f ⊗ 1,g ⊗ 1:A ⊗ kBop\toB ⊗ kBop
b\inB ⊗ kBop
(f ⊗ 1)(a ⊗ z)=b(g ⊗ 1)(a ⊗ z)b-1
a\inA
z\inBop
a=1
1 ⊗ z=b(1 ⊗ z)b-1
Z | |
B ⊗ Bop |
(k ⊗ Bop)=B ⊗ k
b=b' ⊗ 1
z=1
f(a)=b'g(a){b'-1