Jacobson density theorem
Encyclopedia
In mathematics
, more specifically non-commutative ring theory
, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple module
s over a ring R.
The theorem can be applied to show that any primitive ring
can be viewed as a "dense" subring of the ring of linear transformation
s of a vector space. This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson
. This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple
Artinian ring
s.
of U transforming u to v. The natural question now is whether this can be generalized to arbitrary (finite) tuples of elements. More precisely, find necessary and sufficient conditions on the tuple (x1, ..., xn) and (y1, ..., yn) separately, so that there is an element of R with the property that xi·r = yi for all i. If D is the set of all R-module endomorphisms of U, then Schur's lemma
asserts that D is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the x's are linearly independent over D.
With the above in mind, theorem may be stated this way:
The Jacobson Density Theorem
The proof also relies on the following theorem proven in p. 185:
Theorem
Proof (of the Jacobson density theorem)
, and End(DU) is viewed as a subspace of UU and is given the subspace topology
, then R acts densely on U if and only if R is dense set
in End(DU) with this topology.
's conclusion about the structure of simple
right Artinian ring
s is recovered. The Jacobson density theorem also characterizes right or left primitive ring
s as dense subrings of the ring of D-linear transformations on some D- vector space U, where D is a division ring.
, which states that, for a *-algebra A of operators on a Hilbert space
H, the double commutant A′′ can be approximated by A on any given finite set of vectors. See also the Kaplansky density theorem
in the von Neumann algebra setting.
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...
, more specifically non-commutative 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...
, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple module
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
s over a ring R.
The theorem can be applied to show that any primitive ring
Primitive ring
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.- Definition :...
can be viewed as a "dense" subring of the ring of linear transformation
Linear transformation
In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...
s of a vector space. This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson
Nathan Jacobson
Nathan Jacobson was an American mathematician....
. This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple
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...
Artinian ring
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
s.
Motivation and formal statement
Let R be a ring and let U be a simple right R-module. If u is a non-zero element of U, u·R = U (where u·R is the cyclic submodule of U generated by u). Therefore, if u and v are non-zero elements of U, there is an element of R that induces an endomorphismEndomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
of U transforming u to v. The natural question now is whether this can be generalized to arbitrary (finite) tuples of elements. More precisely, find necessary and sufficient conditions on the tuple (x1, ..., xn) and (y1, ..., yn) separately, so that there is an element of R with the property that xi·r = yi for all i. If D is the set of all R-module endomorphisms of U, then Schur's lemma
Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...
asserts that D is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the x's are linearly independent over D.
With the above in mind, theorem may be stated this way:
The Jacobson Density Theorem
- Let U be a simple right R-module and write D = EndR(U). Let A be any D-linear transformation on U and let X be a finite D-linearly independent subset of U. Then there exists an element r of R such that A(x) = x·r for all x in X.
Proof
In the Jacobson density theorem, the right R-module U is simultaneously viewed as a left D-module where D=EndR(U) module in the natural way: the action g·u is defined to be g(u). It can be verified that this is indeed a left module structure on U. As noted before, Schur's lemma proves D is a division ring if U is simple, and so U is a vector space over D.The proof also relies on the following theorem proven in p. 185:
Theorem
- Let U be a simple right R-module and let D = EndR(U) - the set of all R module endomorphisms of U. Let X be a finite subset of U and write I = annR(X) - the annihilatorAnnihilator (ring theory)In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.-Definitions:Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M...
of X in R. Let u be in U with u·I = 0. Then u is in XD; the D-spanLinear spanIn the mathematical subfield of linear algebra, the linear span of a set of vectors in a vector space is the intersection of all subspaces containing that set...
of X.
Proof (of the Jacobson density theorem)
- We proceed by mathematical inductionMathematical inductionMathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers...
on the number n of elements in X. If n=0 so that X is empty, then the theorem is vacuously true and the base case for induction is verified. Now we assume that X is non-empty with cardinality n. Let x be an element of X and write Y = X \ {x}. If A is any D-linear transformation on U, the induction hypothesis guarantees that there exists an s in R such that A(y) = y·s for all y in Y.
- Write I = annR(Y). It is easily seen that x·I is a submodule of U. If it were the case that x·I = 0, then the previous theorem would indicate that x would be in the D-span of Y. This would contradict the linear independence of X, so it must be that x·I ≠ 0. So, by simplicity of U, the submodule x·I = U. Since A(x) - x·s is in U=x·I, there exists i in I such that x·i = A(x) - x·s.
- After defining r = s + i, we compute that y·r = y·(s + i) = y·s + y·i = y·s = A(y) for all y in Y. Also, x·r = x·(s + i) = x·s + A(x) - x·s = A(x). Therefore, A(z) = z·r for all z in X, as desired. This completes the inductive step of the proof. It follows now from mathematical induction that the theorem is true for finite sets X of any size.
Topological characterization
A ring R is said to act densely on a simple right R-module U if it satisfies the conclusion of the Jacobson density theorem. There is a topological reason for describing R as "dense". Firstly, R can be identified with a subring of End(DU) by identifying each element of R with the D linear transformation it induces by right multiplication. If U is given the discrete topology, and if UU is given the product topologyProduct topology
In topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
, and End(DU) is viewed as a subspace of UU and is given the subspace topology
Subspace topology
In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology .- Definition :Given a topological space and a subset S of X, the...
, then R acts densely on U if and only if R is dense set
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
in End(DU) with this topology.
Consequences
The Jacobson density theorem has various important consequences in the structure theory of rings. Notably, the Artin–Wedderburn theoremArtin–Wedderburn theorem
In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
's conclusion about the structure of simple
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...
right Artinian ring
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
s is recovered. The Jacobson density theorem also characterizes right or left primitive ring
Primitive ring
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.- Definition :...
s as dense subrings of the ring of D-linear transformations on some D- vector space U, where D is a division ring.
Relations to other results
This result is related to the Von Neumann bicommutant theoremVon Neumann bicommutant theorem
In mathematics, specifically functional analysis, the von Neumann bicommutant theorem relates the closure of a set of bounded operators on a Hilbert space in certain topologies to the bicommutant of that set. In essence, it is a connection between the algebraic and topological sides of operator...
, which states that, for a *-algebra A of operators on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
H, the double commutant A′′ can be approximated by A on any given finite set of vectors. See also the Kaplansky density theorem
Kaplansky density theorem
In the theory of von Neumann algebras, the Kaplansky density theorem states thatif A is a *-subalgebra of the algebra B of bounded operators on a Hilbert space H, then the strong closure of the unit ball of A in B is the unit ball of the strong closure of A in B...
in the von Neumann algebra setting.