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 Template:Mvar.<ref>Isaacs, p. 184</ref>

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.<ref>Such rings of linear transformations are also known as full linear rings.</ref><ref name="Isaacs187">Isaacs, Corollary 13.16, p. 187</ref> This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson.<ref>Jacobson, Nathan "Structure Theory of Simple Rings Without Finiteness Assumptions"</ref> 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 Template:Mvar be a ring and let Template:Mvar be a simple right Template:Mvar-module. If Template:Mvar is a non-zero element of Template:Mvar, Template:Math (where Template:Math is the cyclic submodule of Template:Mvar generated by Template:Mvar). Therefore, if Template:Mvar are non-zero elements of Template:Mvar, there is an element of Template:Mvar that induces an endomorphism of Template:Mvar transforming Template:Mvar to Template:Mvar. 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 Template:Math and Template:Math separately, so that there is an element of Template:Mvar with the property that Template:Math for all Template:Mvar. If Template:Mvar is the set of all Template:Mvar-module endomorphisms of Template:Mvar, then Schur's lemma asserts that Template:Mvar is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the Template:Mvar are linearly independent over Template:Mvar.

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

The Jacobson density theorem. Let Template:Mvar be a simple right Template:Mvar-module, Template:Math, and Template:Math a finite and Template:Mvar-linearly independent set. If Template:Mvar is a Template:Mvar-linear transformation on Template:Mvar then there exists Template:Math such that Template:Math for all Template:Mvar in Template:Mvar.<ref>Isaacs, Theorem 13.14, p. 185</ref>

Proof

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

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

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

Proof of the Jacobson density theorem

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

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

Define Template:Math and observe that for all Template:Mvar in Template:Mvar we have:

<math>\begin{align}

y \cdot r &= y \cdot(s + i) \\ &= y \cdot s + y \cdot i \\ &= y \cdot s && (\text{since } i\in \text{ann}_R(Y)) \\ &= A(y) \end{align}</math>

Now we do the same calculation for Template:Mvar:

<math>\begin{align}

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{align}</math>

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

Topological characterization

A ring Template:Mvar is said to act densely on a simple right Template:Mvar-module Template:Mvar if it satisfies the conclusion of the Jacobson density theorem.<ref>Herstein, Definition, p. 40</ref> There is a topological reason for describing Template:Mvar as "dense". Firstly, Template:Mvar can be identified with a subring of Template:Math by identifying each element of Template:Mvar with the Template:Mvar linear transformation it induces by right multiplication. If Template:Mvar is given the discrete topology, and if Template:Mvar is given the product topology, and Template:Math is viewed as a subspace of Template:Mvar and is given the subspace topology, then Template:Mvar acts densely on Template:Mvar if and only if Template:Mvar is dense set in Template:Math with this topology.<ref>It turns out this topology is the same as the compact-open topology in this case. Herstein, p. 41 uses this description.</ref>

Consequences

The Jacobson density theorem has various important consequences in the structure theory of rings.<ref>Herstein, p. 41</ref> 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 Template:Mvar-linear transformations on some Template:Mvar-vector space Template:Mvar, where Template:Mvar is a division ring.<ref name="Isaacs187"/>

Relations to other results

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

References

• Template:Cite book
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External links

• PlanetMath page