Skolem–Noether theorem

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.

Contents 1 Statement 2 Proof 3 Notes 4 References 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 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 : A → B, 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 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. Since {displaystyle f(1)=1neq 0} the map f is injective by simplicity of A, so A is also finite-dimensional. Hence 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, {displaystyle Botimes _{k}B^{text{op}}} is a matrix algebra and that {displaystyle Aotimes _{k}B^{text{op}}} is simple. By the first part applied to the maps {displaystyle fotimes 1,gotimes 1:Aotimes _{k}B^{text{op}}to Botimes _{k}B^{text{op}}} , there exists {displaystyle bin Botimes _{k}B^{text{op}}} such that {displaystyle (fotimes 1)(aotimes z)=b(gotimes 1)(aotimes z)b^{-1}} for all {displaystyle ain A} and {displaystyle zin B^{text{op}}} . Taking {displaystyle a=1} , we find {displaystyle 1otimes z=b(1otimes z)b^{-1}} for all z. That is to say, b is in {displaystyle Z_{Botimes B^{text{op}}}(kotimes B^{text{op}})=Botimes k} and so we can write {displaystyle b=b'otimes 1} . Taking {displaystyle z=1} this time we find {displaystyle f(a)=b'g(a){b'^{-1}}} , which is what was sought.

Notes ^ Lorenz (2008) p.173 ^ Farb, Benson; Dennis, R. Keith (1993). Noncommutative Algebra. Springer. ISBN 9780387940571. ^ Gille & Szamuely (2006) p. 40 ^ Lorenz (2008) p. 174 References Skolem, Thoralf (1927). "Zur Theorie der assoziativen Zahlensysteme". Skrifter Oslo (in German) (12): 50. JFM 54.0154.02. A discussion in Chapter IV of Milne, class field theory [1] Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001. Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001. Categories: Theorems in ring theory