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.

In a general formulation, let *A* and *B* be simple unitary 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 *BxB = (x)* 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 of finite dimension. 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.

First suppose
B
=
M
n
(
k
)
=
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
→
V
f
. But such *b* must be an element of
M
n
(
k
)
=
B
. For the general case, note that
B
⊗
B
op
is a matrix algebra and thus by the first part this algebra has an element *b* such that

(
f
⊗
1
)
(
a
⊗
z
)
=
b
(
g
⊗
1
)
(
a
⊗
z
)
b
−
1

for all
a
∈
A
and
z
∈
B
op
. Taking
a
=
1
, we find

1
⊗
z
=
b
(
1
⊗
z
)
b
−
1
for all *z*. That is to say, *b* is in
Z
B
⊗
B
op
(
k
⊗
B
op
)
=
B
⊗
k
and so we can write
b
=
b
′
⊗
1
. Taking
z
=
1
this time we find

f
(
a
)
=
b
′
g
(
a
)
b
′
−
1
,

which is what was sought.