“Skolem–Noether theorem”的意思、由来-开放百科全书

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

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. 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

In particular, every automorphism of a central simple k-algebra is an inner automorphism.^[3]^[4]

Proof

First suppose

. Then f and g define the actions of A on

; let

denote the A-modules thus obtained. Any 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, note that

is a matrix algebra and that

is simple. By the first part applied to the maps

, there exists

such that

for all z. That is to say, b is in

and so we can write

. Taking

this time we find

Notes

1. ^Lorenz (2008) p.173
2. ^{{cite book|last=Farb|first=Benson|title=Noncommutative Algebra|year=1993|publisher=Springer|isbn=9780387940571|author2=Dennis, R. Keith }}
3. ^Gille & Szamuely (2006) p.40
4. ^Lorenz (2008) p.174

References

{{cite journal | jfm=54.0154.02| journal=Skrifter Oslo | year=1927 | number=12 | pages=50 | language=German | first=Thoralf | last= Skolem | authorlink=Thoralf Skolem | title=Zur Theorie der assoziativen Zahlensysteme }}

A discussion in Chapter IV of Milne, class field theory

{{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=Cambridge University Press | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}

{{cite book | first=Falko | last=Lorenz | title=Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics | year=2008 | publisher=Springer | isbn=978-0-387-72487-4 | zbl=1130.12001 }}

2 : Ring theory|Theorems in algebra