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 $B={\mathrm{M}}_n(k)={\mathrm{End}}_k(k^n)$. Then f and g define the actions of A on $k^n$; let $V_f,V_g$ denote the A-modules thus obtained. Since $f(1)=1\ne 0$ the map f is injective by simplicity of A, so A is also finite-dimensional. Hence two simple A-modules are isomorphic and $V_f,V_g$ are finite direct sums of simple A-modules. Since they have the same dimension, it follows that there is an isomorphism of A-modules $b:V_g\to V_f$. But such b must be an element of ${\mathrm{M}}_n(k)=B$. For the general case, $B{\otimes }_k{B}^{\text{op}}$ is a matrix algebra and that $A{\otimes }_k{B}^{\text{op}}$ is simple. By the first part applied to the maps $f\otimes 1,g\otimes 1:A{\otimes }_k{B}^{\text{op}}\to B{\otimes }_k{B}^{\text{op}}$, there exists $b\in B{\otimes }_k{B}^{\text{op}}$ such that

$(f\otimes 1)(a\otimes z)=b(g\otimes 1)(a\otimes z)b^{-1}$

for all $a\in A$ and $z\in B^{\text{op}}$. Taking $a=1$, we find

$1\otimes z=b(1\otimes z)b^{-1}$

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

$f(a)=b^{\prime }g(a)b{\text{'}}^{-1}$,

which is what was sought.

References

• Skolem, Thoralf (1927). "Zur Theorie der assoziativen Zahlensysteme". Skrifter Oslo (in German) (12): 50. JFM 54.0154.02.{{cite journal}}: CS1 maint: unrecognized language (link)
• 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. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
• Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001.