In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the **Jacobson density theorem** is a theorem concerning simple modules over a ring R.[1]

The theorem can be applied to show that any primitive ring can be viewed as a “dense” subring of the ring of linear transformations of a vector space.[2][3] This theorem first appeared in the literature in 1945, in the famous paper “Structure Theory of Simple Rings Without Finiteness Assumptions” by Nathan Jacobson.[4] This can be viewed as a kind of generalization of the Artin-Wedderburn theorem‘s conclusion about the structure of simpleArtinian rings.

## . . . Jacobson density theorem . . .

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, 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 (*x*_{1}, …, *x _{n}*) and (

*y*

_{1}, …,

*y*) separately, so that there is an element of R with the property that

_{n}*x*•

_{i}*r*=

*y*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

_{i}_{i}are linearly independent over D.

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

**The Jacobson Density Theorem.**Let U be a simple right R-module,*D*= End(*U*), and_{R}*X*⊂*U*a finite and D-linearly independent set. If A is a D-linear transformation on U then there exists*r*∈*R*such that*A*(*x*) =*x*•*r*for all x in X.[5]

## . . . Jacobson density theorem . . .

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