Skolem–Noether theorem
Encyclopedia
In mathematics
, the Skolem–Noether theorem, named after Thoralf Skolem
and Emmy Noether
, is an important result in ring theory
which characterizes the automorphism
s of simple ring
s.
The theorem was first published by Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme (German
: On the theory of associative number systems) and later rediscovered by Noether.
since given x nonzero in K, the simplicity of B implies that the nonzero two-sided ideal Bx 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
. 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 Brauer algebra
is inner
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Skolem–Noether theorem, named after Thoralf Skolem
Thoralf Skolem
Thoralf Albert Skolem was a Norwegian mathematician known mainly for his work on mathematical logic and set theory.-Life:...
and Emmy Noether
Emmy Noether
Amalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
, is an important result in ring theory
Ring theory
In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
which characterizes the automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
s of simple ring
Simple ring
In abstract algebra, a simple ring is a non-zero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
s.
The theorem was first published by Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme (German
German language
German is a West Germanic language, related to and classified alongside English and Dutch. With an estimated 90 – 98 million native speakers, German is one of the world's major languages and is the most widely-spoken first language in the European Union....
: On the theory of associative number systems) and later rediscovered by Noether.
Skolem–Noether theorem
In a general formulation, let A and B be simple rings, and let K = Z(B) be the centre of B. Notice that K is a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
since given x nonzero in K, the simplicity of B implies that the nonzero two-sided ideal Bx is the whole of B, and hence that x is a unit
Unit (ring theory)
In mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
. Suppose further that the dimension
Dimension (vector space)
In mathematics, the dimension of a vector space V is the cardinality of a basis of V. It is sometimes called Hamel dimension or algebraic dimension to distinguish it from other types of dimension...
of B over K is finite, i.e. that B is a central simple algebra
Central simple algebra
In ring theory and related areas of mathematics a central simple algebra over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K...
. Then given K-algebra homomorphisms
- f, g : A → B
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
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
of a Brauer algebra
Brauer algebra
In mathematics, a Brauer algebra is an algebra introduced by used in the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the general linear group in Schur–Weyl duality....
is inner
Inner automorphism
In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
.