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.

Statement

In a general formulation, let A and B be simple 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 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.

Proof

First suppose $B=\operatorname {M} _{n}(k)=\operatorname {End} _{k}(k^{n})$. Then f and g define the actions of A on $k^{n}$; let $V_{f},V_{g}$ denote the A-modules thus obtained. Since they have the same dimension, there is an isomorphism $b:V_{g}\to V_{f}$. But such b must be an element of $\operatorname {M} _{n}(k)=B$. For the general case, note that $B\otimes B^{\text{op}}$ is a matrix algebra and thus by the first part this algebra has an element b 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 $b=b'\otimes 1$. Taking $z=1$ this time we find

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

which is what was sought.

References

• Thoralf Skolem, Zur Theorie der assoziativen Zahlensysteme, 1927
• A discussion in Chapter IV of Milne, class field theory [1]