Math 2.0

Associative rings and algebras Primitive and prime rings Theorem

Jacobson density theorem

Builds on RingSimple moduleEndomorphism ringEndomorphism ring of a simple module is a division ringDivision ring

Statement

Let RR be a ring with identity and let UU be a simple left RR-module. Let D=End⁡R(U)D = \End_R(U) be its endomorphism ring, which is a division ring, and view UU as a vector space over DD. Let u1,…,un∈Uu_1, \ldots, u_n \in U be linearly independent over DD, and let v1,…,vn∈Uv_1, \ldots, v_n \in U be arbitrary. Then there is r∈Rr \in R with rui=vir u_i = v_i for i=1,…,ni = 1, \ldots, n.

Remarks

DD is a division ring by Schur's lemma. The proof is by induction on nn; the key step shows that if W⊆UW \subseteq U is a finite-dimensional DD-subspace and u∉Wu \notin W, then some r∈Rr \in R kills WW but not uu. Example: let R=Mn(k)R = M_n(k) for a field kk and U=knU = k^n, so that D=kD = k. The theorem says that any nn linearly independent vectors can be sent to any nn vectors by a matrix. If UU is finite-dimensional over DD, the map R→End⁡D(U)R \to \End_D(U) is surjective; when RR is a finite-dimensional algebra over an algebraically closed field this gives Burnside's theorem.

Going deeper

For a student

The theorem is the main tool for the structure of primitive rings: a left primitive ring is isomorphic to a dense subring of End⁡D(U)\End_D(U). If in addition RR is left Artinian, the subring is all of End⁡D(U)\End_D(U) with UU finite-dimensional, so R≅Mn(D)R \cong M_n(D). This is the route to Wedderburn's theorem on simple Artinian rings and to the Artin–Wedderburn theorem.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Division ring, concept, no evidence levelDivision ringconceptEndomorphism ring, concept, no evidence levelEndomorphism ringconceptGroup, concept, no evidence levelGroupconceptModule over a ring, definition, no evidence levelModule over a ringdefinitionSimple module, concept, no evidence levelSimple moduleconceptRing, definition, no evidence levelRingdefinitionSchur's lemma, statement, no evidence levelSchur's lemmastatementUnit of a ring, definition, no evidence levelUnit of a ringdefinitionEndomorphism ring of a simple module is a division ring, statement, no evidence levelEndomorphism ring of …statementDivision ring, definition, no evidence levelDivision ringdefinitionJacobson density theorem, statement, no evidence levelJacobson density theo…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Division ring concept
  2. Endomorphism ring concept
  3. Group concept
  4. Module over a ring definition
  5. Simple module concept
  6. Ring definition
  7. Schur's lemma statement
  8. Unit of a ring definition
  9. Endomorphism ring of a simple module is a division ring statement
  10. Division ring definition
  11. Jacobson density theorem statement

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Jacobson_density_theorem
About this page
TagEK0T7FRKCY
URIm2://EK0T7FRKCY
Kindstatement, theorem
WritingW0 stub
MSC16D60
StatusThis copy of the site has no build data. Run m2 status EK0T7FRKCY.
AboutPrimitive ring, Simple module
UsesRing, Simple module, Endomorphism ring, Endomorphism ring of a simple module is a division ring, Division ring
Same as
Changed2 November 2026

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