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 for a ring, the image of the ring in its bicommutant is dense in it.
More precisely, treat the irreducible module as a vector space over the division ring of all its endomorphisms as abelian group that commute with the ring action. That this 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 tuple of elements of the same length, there exists a ring element whose action transforms the first tuple to the second.
In particular, when the ring is primitive then it is a dense subring of linear transformations of a vector space over a division ring.