Jacobson Density Theorem - Motivation and Formal Statement

Motivation and Formal Statement

Let R be a ring and let U be a simple right R-module. If u is a non-zero element of U, u·R = U (where u·R is the cyclic submodule of U generated by u). Therefore, if u and v are non-zero elements of U, there is an element of R that induces an endomorphism of U transforming u to v. The natural question now is whether this can be generalized to arbitrary (finite) tuples of elements. More precisely, find necessary and sufficient conditions on the tuple (x₁, ..., x_n) and (y₁, ..., y_n) separately, so that there is an element of R with the property that x_i·r = y_i for all i. If D is the set of all R-module endomorphisms of U, then Schur's lemma asserts that D is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the x's are linearly independent over D.

With the above in mind, theorem may be stated this way:

The Jacobson Density Theorem

Let U be a simple right R-module and write D = End(U_R). Let A be any D-linear transformation on U and let X be a finite D-linearly independent subset of U. Then there exists an element r of R such that A(x) = x·r for all x in X.