Monotonicity of entailment

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 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 automorphism 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. Any 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 $\operatorname {M} _{n}(k)=B$. 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

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