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, the image of R in its bicommutant is dense in it.

More precisely, treat M as a vector space over the division ring D of all its endomorphisms as abelian group that commute with the action of R. (That D is a division ring is a consequence of Schur's lemma.) Then, given any ordered tuple of elements in the module that are linearly independent over the division ring, 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. See also the Kaplansky density theorem in the von Neumann algebra setting.