[2][3] This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson.
[4] This can be viewed as a kind of generalization of the Artin-Wedderburn theorem's conclusion about the structure of simple Artinian rings.
More precisely, find necessary and sufficient conditions on the tuple (x1, ..., xn) and (y1, ..., yn) separately, so that there is an element of R with the property that xi • r = yi for all i.
[6] As noted before, Schur's lemma proves D is a division ring if U is simple, and so U is a vector space over D. The proof also relies on the following theorem proven in (Isaacs 1993) p. 185: We use induction on |X|.
A ring R is said to act densely on a simple right R-module U if it satisfies the conclusion of the Jacobson density theorem.
[3] 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 approximated by A on any given finite set of vectors.