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
Associated With
Has Statement Form
In Branch
Principal Result Of
Source Jacobson density theorem (Wikipedia)
Proved By
Source Jacobson density theorem (Wikipedia)
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
Proved By: Nathan Jacobson, Lead sentence
This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson.
Principal Result Of: Nathan Jacobson, Lead sentence of the article on the result
This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by Nathan Jacobson.
In Group: Ring Theory and Commutative Algebra, Lead section
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson
View the SourceReader Challenges (0)
No disputes yet. Spotted an error or a better source? Open the first one.
Sign in to dispute this or suggest a correction.