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In ring theory, a branch of mathematics, the Skolem–Noether theorem characterizes the
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
s of
simple ring In abstract algebra, a branch of mathematics, a simple ring is a non-zero ring that has no two-sided ideal besides the zero ideal and itself. In particular, a commutative ring is a simple ring if and only if it is a field. The center of a sim ...
s. It is a fundamental result in the theory of
central simple algebra In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' that is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple ...
s. The theorem was first published by
Thoralf Skolem Thoralf Albert Skolem (; 23 May 1887 – 23 March 1963) was a Norwegian mathematician who worked in mathematical logic and set theory. Life Although Skolem's father was a primary school teacher, most of his extended family were farmers. Skole ...
in 1927 in his paper ''Zur Theorie der assoziativen Zahlensysteme'' (
German German(s) may refer to: * Germany, the country of the Germans and German things **Germania (Roman era) * Germans, citizens of Germany, people of German ancestry, or native speakers of the German language ** For citizenship in Germany, see also Ge ...
: ''On the theory of associative number systems'') and later rediscovered by
Emmy Noether Amalie Emmy Noether (23 March 1882 – 14 April 1935) was a German mathematician who made many important contributions to abstract algebra. She also proved Noether's theorem, Noether's first and Noether's second theorem, second theorems, which ...
.


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 since given ''x'' nonzero in ''k'', the simplicity of ''B'' implies that the nonzero two-sided ideal is the whole of ''B'', and hence that ''x'' is a
unit Unit may refer to: General measurement * Unit of measurement, a definite magnitude of a physical quantity, defined and adopted by convention or by law **International System of Units (SI), modern form of the metric system **English units, histo ...
. If the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
of ''B'' over ''k'' is finite, i.e. if ''B'' is a
central simple algebra In ring theory and related areas of mathematics a central simple algebra (CSA) over a field ''K'' is a finite-dimensional associative ''K''-algebra ''A'' that is simple, and for which the center is exactly ''K''. (Note that ''not'' every simple ...
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'' :''g''(''a'') = ''b'' · ''f''(''a'') · ''b''−1. In particular, every
automorphism In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphism ...
of a central simple ''k''-algebra is an
inner automorphism In abstract algebra, an inner automorphism is an automorphism of a group, ring, or algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within thos ...
.Gille & Szamuely (2006) p. 40Lorenz (2008) p. 174


Proof

First suppose B = \operatorname_n(k) = \operatorname_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 f(1) = 1 \neq 0 the map ''f'' is injective by simplicity of ''A'', so ''A'' is also finite-dimensional. Hence 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_n(k) = B. For the general case, B \otimes_k B^ is a matrix algebra and that A \otimes_k B^ is simple. By the first part applied to the maps f \otimes 1, g \otimes1 : A \otimes_k B^ \to B \otimes_k B^, there exists b \in B \otimes_k B^ such that :(f \otimes 1)(a \otimes z) = b (g \otimes 1)(a \otimes z) b^ for all a \in A and z \in B^. Taking a = 1, we find :1 \otimes z = b (1\otimes z) b^ for all ''z''. That is to say, ''b'' is in Z_(k \otimes B^) = B \otimes k and so we can write b = b' \otimes 1. Taking z = 1 this time we find :f(a)= b' g(a) , which is what was sought.


Notes


References

* * A discussion in Chapter IV of Milne, class field theor

* * {{DEFAULTSORT:Skolem-Noether theorem Theorems in ring theory