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 let K = Z(B) 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 Bx 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. 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 endomorphism of a central simple k-algebra is an inner automorphism.

References

• Thoralf Skolem, Zur Theorie der assoziativen Zahlensysteme, 1927
• A proof [1]
• A discussion in Chapter IV of [2]