Skolem–Noether theorem

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.

Statement

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

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 [math]\displaystyle{ B = \operatorname{M}_n(k) = \operatorname{End}_k(k^n) }[/math]. Then f and g define the actions of A on [math]\displaystyle{ k^n }[/math]; let [math]\displaystyle{ V_f, V_g }[/math] denote the A-modules thus obtained. Any two simple A-modules are isomorphic and [math]\displaystyle{ V_f, V_g }[/math] are finite direct sums of simple A-modules. Since they have the same dimension, it follows that there is an isomorphism of A-modules [math]\displaystyle{ b: V_g \to V_f }[/math]. But such b must be an element of [math]\displaystyle{ \operatorname{M}_n(k) = B }[/math]. For the general case, [math]\displaystyle{ B \otimes_k B^{\text{op}} }[/math] is a matrix algebra and that [math]\displaystyle{ A \otimes_k B^{\text{op}} }[/math] is simple. By the first part applied to the maps [math]\displaystyle{ f \otimes 1, g \otimes1 : A \otimes_k B^{\text{op}} \to B \otimes_k B^{\text{op}} }[/math], there exists [math]\displaystyle{ b \in B \otimes_k B^{\text{op}} }[/math] such that

[math]\displaystyle{ (f \otimes 1)(a \otimes z) = b (g \otimes 1)(a \otimes z) b^{-1} }[/math]

for all [math]\displaystyle{ a \in A }[/math] and [math]\displaystyle{ z \in B^{\text{op}} }[/math]. Taking [math]\displaystyle{ a = 1 }[/math], we find

[math]\displaystyle{ 1 \otimes z = b (1\otimes z) b^{-1} }[/math]

for all z. That is to say, b is in [math]\displaystyle{ Z_{B \otimes B^{\text{op}}}(k \otimes B^{\text{op}}) = B \otimes k }[/math] and so we can write [math]\displaystyle{ b = b' \otimes 1 }[/math]. Taking [math]\displaystyle{ z = 1 }[/math] this time we find

[math]\displaystyle{ f(a)= b' g(a) {b'^{-1}} }[/math],

which is what was sought.

References

• Skolem, Thoralf (1927). "Zur Theorie der assoziativen Zahlensysteme" (in German). Skrifter Oslo (12): 50. 
• A discussion in Chapter IV of Milne, class field theory [1]
• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. 
• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4.