# 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 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*^{[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 . Then *f* and *g* define the actions of *A* on ; let denote the *A*-modules thus obtained. Any two simple *A*-modules are isomorphic and are finite direct sums of simple *A*-modules. Since they have the same dimension, it follows that there is an isomorphism of *A*-modules . But such *b* must be an element of . For the general case, note that is a matrix algebra and thus by the first part this algebra has an element *b* such that

for all and . Taking , we find

for all *z*. That is to say, *b* is in and so we can write . Taking this time we find

which is what was sought.

## Notes

## References

- {{#invoke:Citation/CS1|citation

|CitationClass=journal }}

|CitationClass=book }}

- {{#invoke:citation/CS1|citation

|CitationClass=book }}