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.
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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.
Statement
In a general formulation, let A and B be simple unitary rings, and let k be the centre of B. Notice that 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. Suppose further that the dimension of B over k is finite, i.e. that B is a central simple algebra of finite dimension. Then given k-algebra homomorphisms
f, g : A → B,there exists a unit b in B such that for all a in A
g(a) = b · f(a) · b−1.In particular, every automorphism of a central simple k-algebra is an inner automorphism.
Proof
First suppose
for all
for all z. That is to say, b is in
which is what was sought.