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 simple Artinian rings.

Contents

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, ..., xn) and (y1, ..., yn) separately, so that there is an element of R with the property that xi·r = 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 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 = EndR(U). 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.[5]

Proof

In the Jacobson density theorem, the right R-module U is simultaneously viewed as a left D-module where D=EndR(U) module in the natural way: the action g·u is defined to be g(u). It can be verified that this is indeed a left module structure on U.[6] As noted before, Schur's lemma proves D is a division ring if U is simple, and so U is a vector space over D.

The proof also relies on the following theorem proven in (Isaacs 1993) p. 185:

Theorem

Let U be a simple right R-module and let D = EndR(U) - the set of all R module endomorphisms of U. Let X be a finite subset of U and write I = annR(X) - the annihilator of X in R. Let u be in U with u·I = 0. Then u is in XD; the D-span of X.

Proof (of the Jacobson density theorem)

We proceed by mathematical induction on the number n of elements in X. If n=0 so that X is empty, then the theorem is vacuously true and the base case for induction is verified. Now we assume that X is non-empty with cardinality n. Let x be an element of X and write Y = X \ {x}. If A is any D-linear transformation on U, the induction hypothesis guarantees that there exists an s in R such that A(y) = y·s for all y in Y.
Write I = annR(Y). It is easily seen that x·I is a submodule of U. If it were the case that x·I = 0, then the previous theorem would indicate that x would be in the D-span of Y. This would contradict the linear independence of X, so it must be that x·I ≠ 0. So, by simplicity of U, the submodule x·I = U. Since A(x) - x·s is in U=x·I, there exists i in I such that x·i = A(x) - x·s.
After defining r = s + i, we compute that y·r = y·(s + i) = y·s + y·i = y·s = A(y) for all y in Y.[7] Also, x·r = x·(s + i) = x·s + A(x) - x·s = A(x). Therefore, A(z) = z·r for all z in X, as desired. This completes the inductive step of the proof. It follows now from mathematical induction that the theorem is true for finite sets X of any size.

Topological characterization

A ring R is said to act densely on a simple right R-module U if it satisfies the conclusion of the Jacobson density theorem.[8] There is a topological reason for describing R as "dense". Firstly, R can be identified with a subring of End(DU) by identifying each element of R with the D linear transformation it induces by right multiplication. If U is given the discrete topology, and if UU is given the product topology, and End(DU) is viewed as a subspace of UU and is given the subspace topology, then R acts densely on U if and only if R is dense set in End(DU) with this topology[9].

Consequences

The Jacobson density theorem has various important consequences in the structure theory of rings.[10] Notably, the Artin–Wedderburn theorem's conclusion about the structure of simple right Artinian rings is recovered. The Jacobson density theorem also characterizes right or left primitive rings as dense subrings of the ring of D-linear transformations on some D- vector space U, where D is a division ring.[11]

Relations to other results

This result is related to the Von Neumann bicommutant theorem, which states that, for a *-algebra A of operators on a Hilbert space H, the double commutant A′′ can be approximated by A on any given finite set of vectors. See also the Kaplansky density theorem in the von Neumann algebra setting.

Notes

  1. ^ Isaacs, p. 184
  2. ^ Such rings of linear transformations are also known as full linear rings.
  3. ^ Isaacs, Corollary 13.16, p. 187
  4. ^ Jacobson, Nathan "Structure Theory of Simple Rings Without Finiteness Assumptions"
  5. ^ Isaacs, Theorem 13.14, p. 185
  6. ^ Incidentally it is also a D-R bimodule structure.
  7. ^ Of course, y·i=0 by definition of I.
  8. ^ Herstein, Definition, p. 40
  9. ^ It turns out this topology is the same as the compact-open topology in this case. Herstein, p. 41 uses this description.
  10. ^ Herstein, p. 41
  11. ^ Isaacs, Corollary 13.16, p. 187

References

External links