Editing Skolem–Noether theorem

{{Short description|Theorem characterizing the automorphisms of simple rings}}
In [[ring theory]], a branch of mathematics, the '''Skolem–Noether theorem''' characterizes the [[automorphism]]s of [[simple ring]]s. It is a fundamental result in the theory of [[central simple algebra]]s.

The theorem was first published by [[Thoralf Skolem]] in 1927 in his paper ''Zur Theorie der assoziativen Zahlensysteme'' ([[German language|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 (mathematics)|field]] since given ''x'' nonzero in ''k'', the simplicity of ''B'' implies that the nonzero two-sided ideal {{nowrap|1=''BxB'' = (''x'')}} is the whole of ''B'', and hence that ''x'' is a [[Unit (ring theory)|unit]]. If the [[dimension (vector space)|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''<ref>Lorenz (2008) p.173</ref><ref>{{cite book|last=Farb|first=Benson|title=Noncommutative Algebra|year=1993|publisher=Springer|isbn=9780387940571|author2=Dennis, R. Keith }}</ref>

:''g''(''a'') = ''b'' · ''f''(''a'') · ''b''<sup>−1</sup>.

In particular, every [[automorphism]] of a central simple ''k''-algebra is an [[inner automorphism]].<ref name=GS40>Gille & Szamuely (2006) p. 40</ref><ref name=Lor174>Lorenz (2008) p. 174</ref>

== Proof ==
First suppose <math>B = \operatorname{M}_n(k) = \operatorname{End}_k(k^n)</math>. Then ''f'' and ''g'' define the actions of ''A'' on <math>k^n</math>; let <math>V_f, V_g</math> denote the ''A''-modules thus obtained. Since <math>f(1) = 1 \neq 0 </math> the map ''f'' is injective by simplicity of ''A'', so ''A'' is also finite-dimensional. Hence two simple ''A''-modules are isomorphic and <math>V_f, V_g</math> are finite direct sums of simple ''A''-modules. Since they have the same dimension, it follows that there is an isomorphism of ''A''-modules <math>b: V_g \to V_f</math>. But such ''b'' must be an element of <math>\operatorname{M}_n(k) = B</math>. For the general case, <math>B \otimes_k B^{\text{op}}</math> is a matrix algebra and that <math>A \otimes_k B^{\text{op}}</math> is simple. By the first part applied to the maps <math>f \otimes 1, g \otimes1 : A \otimes_k B^{\text{op}} \to B \otimes_k B^{\text{op}}</math>, there exists <math>b \in B \otimes_k B^{\text{op}}</math> such that
:<math>(f \otimes 1)(a \otimes z) = b (g \otimes 1)(a \otimes z) b^{-1}</math>
for all <math>a \in A</math> and <math>z \in B^{\text{op}}</math>. Taking <math>a = 1</math>, we find
:<math>1 \otimes z = b (1\otimes z) b^{-1}</math>
for all ''z''. That is to say, ''b'' is in <math>Z_{B \otimes B^{\text{op}}}(k \otimes B^{\text{op}}) = B \otimes k</math> and so we can write <math>b = b' \otimes 1</math>. Taking <math>z = 1</math> this time we find
:<math>f(a)= b' g(a) {b'^{-1}}</math>,
which is what was sought.

== Notes ==
{{reflist}}

== References ==
* {{cite journal | jfm=54.0154.02| journal=Skrifter Oslo | year=1927 | number=12 | pages=50 | language=German | first=Thoralf | last= Skolem | authorlink=Thoralf Skolem | title=Zur Theorie der assoziativen Zahlensysteme }}
* A discussion in Chapter IV of [[James Milne (mathematician)|Milne]], class field theory [http://jmilne.org/math/CourseNotes/cft.html]
* {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}
* {{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=Springer | isbn=978-0-387-72487-4 | zbl=1130.12001 }}

{{DEFAULTSORT:Skolem-Noether theorem}}
[[Category:Theorems in ring theory]]