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.

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

there exists a unit b in B such that for all a in A

In particular, every automorphism of a central simple k-algebra is an inner automorphism.

First suppose ${\displaystyle B=\operatorname {M} _{n}(k)=\operatorname {End} _{k}(k^{n})}$. Then f and g define the actions of A on ${\displaystyle k^{n}}$; let ${\displaystyle V_{f},V_{g}}$ denote the A-modules thus obtained. Any two simple A-modules are isomorphic and ${\displaystyle V_{f},V_{g}}$ are finite direct sums of simple A-modules. Since they have the same dimension, it follows that there is an isomorphism of A-modules ${\displaystyle b:V_{g}\to V_{f}}$. But such b must be an element of ${\displaystyle \operatorname {M} _{n}(k)=B}$. For the general case, note that ${\displaystyle B\otimes B^{\text{op}}}$ is a matrix algebra and thus by the first part this algebra has an element b such that

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