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^[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]

Proof

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

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

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

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

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

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

which is what was sought.

Notes

References

• Skolem, Thoralf (1927). "Zur Theorie der assoziativen Zahlensysteme". Skrifter Oslo (in German) (12): 50. JFM 54.0154.02.
• 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.