Jacobson density theorem

Jacobson density theorem

In mathematics, the Jacobson density theorem in ring theory is an important generalization of the Artin-Wedderburn theorem. It is named for Nathan Jacobson.

It states that given any irreducible module "M" for a ring "R", "R" is dense in its bicommutant.

More precisely, let "D" be the ring of all endomorphisms of "M" that commute with the action of "R". By Schur's lemma, "D" is a division ring. Treat "M" as a vector space over the division ring "D". Then, given any ordered tuple of elements in "M" that are linearly independent over "D", and any other similar tuple of elements of the same length, there exists "r" in "R" whose action transforms the first tuple to the second.

In particular, when "R" is a primitive ring, then it is isomorphic with a dense subring of linear transformations of a vector space over a division ring.

This result is related to the Von Neumann bicommutant theorem, which states that, for a *-algebra "A" of operators on a Hilbert space "H", the double commutant A′′ can be approximate by "A" on any given finite set of vectors. See also the Kaplansky density theorem in the von Neumann algebra setting.

References

External links

* [http://planetmath.org/encyclopedia/JacobsonDensityTheorem.html PlanetMath page]


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