Jacobson density theorem

In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring .

The theorem can be applied to show that any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vector space.Isaacs, Corollary 13.16, p. 187 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 rings.

Motivation and formal statement

Let be a ring and let be a simple right -module. If is a non-zero element of , (where is the cyclic submodule of generated by ). Therefore, if are non-zero elements of , there is an element of that induces an endomorphism of transforming to . 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 and separately, so that there is an element of with the property that for all . If is the set of all -module endomorphisms of , then Schur's lemma asserts that is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the are linearly independent over .

With the above in mind, the theorem may be stated this way:

:The Jacobson density theorem. Let be a simple right -module, , and a finite and -linearly independent set. If is a -linear transformation on then there exists such that for all in .

Proof

In the Jacobson density theorem, the right -module is simultaneously viewed as a left -module where , in the natural way: . It can be verified that this is indeed a left module structure on . As noted before, Schur's lemma proves is a division ring if is simple, and so is a vector space over .

The proof also relies on the following theorem proven in p. 185:

:Theorem. Let be a simple right -module, , and a finite set. Write for the annihilator of in . Let be in with . Then is in ; the -span of .

Proof of the Jacobson density theorem

We use induction on . If is empty, then the theorem is vacuously true and the base case for induction is verified.

Assume is non-empty, let be an element of and write If is any -linear transformation on , by the induction hypothesis there exists such that for all in . Write . It is easily seen that is a submodule of . If , then the previous theorem implies that would be in the -span of , contradicting the -linear independence of , therefore . Since is simple, we have: . Since , there exists in such that .

Define and observe that for all in we have:

:$\begin y \cdot r &= y \cdot(s + i) \\ &= y \cdot s + y \cdot i \\ &= y \cdot s && (\text i\in \text_R(Y)) \\ &= A(y) \end$

Now we do the same calculation for :

:$\begin x \cdot r &= x \cdot (s + i) \\ &= x \cdot s + x \cdot i \\ &= x \cdot s + \left ( A(x) - x \cdot s \right )\\ &= A(x) \end$

Therefore, for all in , as desired. This completes the inductive step of the proof. It follows now from mathematical induction that the theorem is true for finite sets of any size.

Topological characterization

A ring is said to act densely on a simple right -module if it satisfies the conclusion of the Jacobson density theorem. There is a topological reason for describing as "dense". Firstly, can be identified with a subring of by identifying each element of with the linear transformation it induces by right multiplication. If is given the discrete topology, and if is given the product topology, and is viewed as a subspace of and is given the subspace topology, then acts densely on if and only if is dense set in with this topology.

Consequences

The Jacobson density theorem has various important consequences in the structure theory of rings.Herstein, p. 41 Notably, the Artin–Wedderburn theorem's conclusion about the structure of simple right Artinian rings is recovered. The Jacobson density theorem also characterizes right or left primitive rings as dense subrings of the ring of -linear transformations on some -vector space , where is a division ring.

Relations to other results

This result is related to the Von Neumann bicommutant theorem, which states that, for a *-algebra of operators on a Hilbert space , the double commutant can be approximated by on any given finite set of vectors. In other words, the double commutant is the closure of in the weak operator topology. See also the Kaplansky density theorem in the von Neumann algebra setting.

Notes

References

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External links

PlanetMath page

{{DEFAULTSORT:Jacobson Density Theorem
Theorems in ring theory
Module theory
Articles containing proofs