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

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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 BxB = (x) is the whole of B, and hence that x is a unit. If the 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 : AB,

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

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First suppose  . Then f and g define the actions of A on  ; let   denote the A-modules thus obtained. Since   the map f is injective by simplicity of A, so A is also finite-dimensional. Hence 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,   is a matrix algebra and that   is simple. By the first part applied to the maps  , there exists   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

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  1. ^ Lorenz (2008) p.173
  2. ^ Farb, Benson; Dennis, R. Keith (1993). Noncommutative Algebra. Springer. ISBN 9780387940571.
  3. ^ Gille & Szamuely (2006) p. 40
  4. ^ Lorenz (2008) p. 174

References

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