Jacobson density theorem
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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.
