The Jacobson Density Theorem is a result of noncommutative ring theory concerning simple modules over a ring, showing that any primitive ring can be viewed as a dense subring of the ring of linear transformations of a vector space. First proved by Nathan Jacobson in his 1945 paper on the structure theory of simple rings without finiteness assumptions, it generalizes the conclusion the Artin-Wedderburn Theorem gives for simple Artinian rings to primitive rings that need not satisfy any finiteness condition.
Facts
StatementLet U be a simple right R-module, D = End(U_R), and X a finite D-linearly independent subset of U. Then for any D-linear transformation A on U there is r in R with A(x) = x r for all x in X; consequently any primitive ring is a dense subring of the ring of linear transformations of a vector space. 1 Classification
Statement Form Connections
Sources
1. Jacobson density theorem (Wikipedia)
Introduction, paragraph 2, sentence 1
any primitive ring can be viewed as a "dense" subring of the ring of linear transformations of a vector space
Introduction, paragraph 2, sentence 2
This theorem first appeared in the literature in 1945
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