Jacobson density theorem

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.

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Let R be a ring and let U be a simple right R-module. If u is a non-zero element of U, uR = U (where uR 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 (x1, …, xn) and (y1, …, yn) separately, so that there is an element of R with the property that xir = yi 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 xi 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(UR), and XU a finite and D-linearly independent set. If A is a D-linear transformation on U then there exists rR such that A(x) = xr for all x in X.[5]

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. . . Jacobson density theorem . . .

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