Theorem

Jacobson Density Theorem

Algebra

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

Statement

Let 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

Proof Year

1945 1

Classification

Statement Form

Identity or Equation 1

Connections

Associated With

Ring (Abstract Algebra)

Has Statement Form

Equation

Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.

Identity

In Branch

Ring Theory

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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Jacobson Density Theorem

Associated With

Has Statement Form

In Branch