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Skolem–Noether theorem

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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.

Contents

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 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 : AB,

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.

Proof

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.

References

Skolem–Noether theorem Wikipedia


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