Editing Jacobson density theorem

In [[mathematics]], more specifically non-commutative [[ring theory]], [[Abstract algebra|modern algebra]], and [[module theory]], the '''Jacobson density theorem''' is a theorem concerning [[simple module]]s over a ring {{mvar|R}}.<ref>Isaacs, p. 184</ref>

The theorem can be applied to show that any [[primitive ring]] can be viewed as a "dense" subring of the ring of [[linear transformation]]s of a vector space.<ref>Such rings of linear transformations are also known as [[full linear ring]]s.</ref><ref name="Isaacs187">Isaacs, Corollary 13.16, p. 187</ref> This theorem first appeared in the literature in 1945, in the famous paper "Structure Theory of Simple Rings Without Finiteness Assumptions" by [[Nathan Jacobson]].<ref>[https://www.jstor.org/pss/1990204 Jacobson, Nathan "Structure Theory of Simple Rings Without Finiteness Assumptions"]</ref> This can be viewed as a kind of generalization of the [[Artin-Wedderburn theorem]]'s conclusion about the structure of [[simple ring|simple]] [[Artinian ring]]s.

==Motivation and formal statement==
Let {{mvar|R}} be a ring and let {{mvar|U}} be a simple right {{mvar|R}}-module. If {{mvar|u}} is a non-zero element of {{mvar|U}}, {{math|''u'' • ''R'' {{=}} ''U''}} (where {{math|''u'' • ''R''}} is the cyclic submodule of {{mvar|U}} generated by {{mvar|u}}). Therefore, if {{mvar|u, v}} are non-zero elements of {{mvar|U}}, there is an element of {{mvar|R}} that induces an [[endomorphism]] of {{mvar|U}} transforming {{mvar|u}} to {{mvar|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 {{math|(''x''<sub>1</sub>, ..., ''x<sub>n</sub>'')}} and {{math|(''y''<sub>1</sub>, ..., ''y<sub>n</sub>'')}} separately, so that there is an element of {{mvar|R}} with the property that {{math|''x<sub>i</sub>'' • ''r'' {{=}} ''y<sub>i</sub>''}} for all {{mvar|i}}. If {{mvar|D}} is the set of all {{mvar|R}}-module endomorphisms of {{mvar|U}}, then [[Schur's lemma]] asserts that {{mvar|D}} is a division ring, and the Jacobson density theorem answers the question on tuples in the affirmative, provided that the {{mvar|x<sub>i</sub>}} are linearly independent over {{mvar|D}}.

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

:'''The Jacobson density theorem.''' Let {{mvar|U}} be a simple right {{mvar|R}}-module, {{math|''D'' {{=}} End(''U<sub>R</sub>'')}}, and {{math|''X'' ⊂ ''U''}} a finite and {{mvar|D}}-linearly independent set. If {{mvar|A}} is a {{mvar|D}}-linear transformation on {{mvar|U}} then there exists {{math|''r'' ∈ ''R''}} such that {{math|''A''(''x'') {{=}} ''x'' • ''r''}} for all {{mvar|x}} in {{mvar|X}}.<ref>Isaacs, Theorem 13.14, p. 185</ref>

==Proof==
In the Jacobson density theorem, the right {{mvar|R}}-module {{mvar|U}} is simultaneously viewed as a left {{mvar|D}}-module where {{math|''D'' {{=}} End(''U<sub>R</sub>'')}}, in the natural way: {{math|''g'' • ''u'' {{=}} ''g''(''u'')}}. It can be verified that this is indeed a left module structure on {{mvar|U}}.<ref>Incidentally it is also a {{math|''D''-''R''}} [[bimodule]] structure.</ref> As noted before, Schur's lemma proves {{mvar|D}} is a division ring if {{mvar|U}} is simple, and so {{mvar|U}} is a vector space over {{mvar|D}}.

The proof also relies on the following theorem proven in {{harv|Isaacs|1993}} p.&nbsp;185:

:'''Theorem.''' Let {{mvar|U}} be a simple right {{mvar|R}}-module, {{math|''D'' {{=}} End(''U<sub>R</sub>'')}}, and {{math|''X'' ⊂ ''U''}} a finite set. Write {{math|''I'' {{=}} ann<sub>''R''</sub>(''X'')}} for the [[Annihilator (ring theory)|annihilator]] of {{mvar|X}} in {{mvar|R}}. Let {{mvar|u}} be in {{mvar|U}} with {{math|''u'' • ''I'' {{=}} 0}}. Then {{mvar|u}} is in {{mvar|XD}}; the {{mvar|D}}-[[linear span|span]] of {{mvar|X}}.

===Proof of the Jacobson density theorem=== 
We use [[mathematical induction|induction]] on {{math|{{!}}''X''{{!}}}}. If {{mvar|X}} is empty, then the theorem is vacuously true and the base case for induction is verified.

Assume {{mvar|X}} is non-empty, let {{mvar|x}} be an element of {{mvar|X}} and write {{math|''Y'' {{=}} ''X''&thinsp;\{''x''}.}} If {{mvar|A}} is any {{mvar|D}}-linear transformation on {{mvar|U}}, by the induction hypothesis there exists {{math|''s'' ∈ ''R''}} such that {{math|''A''(''y'') {{=}} ''y'' • ''s''}} for all {{mvar|y}} in {{mvar|Y}}. Write {{math|''I'' {{=}} ann<sub>''R''</sub>(''Y'')}}. It is easily seen that {{math|''x'' • ''I''}} is a submodule of {{mvar|U}}. If {{math|''x'' • ''I'' {{=}} 0}}, then the previous theorem implies that {{mvar|x}} would be in the {{mvar|D}}-span of {{mvar|Y}}, contradicting the {{mvar|D}}-linear independence of {{mvar|X}}, therefore {{math|''x'' • ''I'' ≠ 0}}. Since {{mvar|U}} is simple, we have: {{math|''x'' • ''I'' {{=}} ''U''}}. Since {{math|''A''(''x'') − ''x'' • ''s'' ∈ ''U'' {{=}} ''x'' • ''I''}}, there exists {{mvar|i}} in {{mvar|I}} such that {{math|''x'' • ''i'' {{=}} ''A''(''x'') − ''x'' • ''s''}}.

Define {{math|''r'' {{=}} ''s'' + ''i''}} and observe that for all {{mvar|y}} in {{mvar|Y}} we have:

:<math>\begin{align}
y \cdot r &= y \cdot(s + i) \\
&= y \cdot s + y \cdot i \\
&= y \cdot s && (\text{since } i\in \text{ann}_R(Y)) \\
&= A(y)
\end{align}</math>

Now we do the same calculation for {{mvar|x}}:

:<math>\begin{align}
x \cdot r &= x \cdot (s + i) \\
&= x \cdot s + x \cdot i \\
&= x \cdot s + \left ( A(x) - x \cdot s \right )\\
&= A(x)
\end{align}</math>

Therefore, {{math|''A''(''z'') {{=}} ''z'' • ''r''}} for all {{mvar|z}} in {{mvar|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 {{mvar|X}} of any size.

==Topological characterization==
A ring {{mvar|R}} is said to '''act densely''' on a simple right {{mvar|R}}-module {{mvar|U}} if it satisfies the conclusion of the Jacobson density theorem.<ref>Herstein, Definition, p. 40</ref> There is a topological reason for describing {{mvar|R}} as "dense". Firstly, {{mvar|R}} can be identified with a subring of {{math|End(''<sub>D</sub>U'')}} by identifying each element of {{mvar|R}} with the {{mvar|D}} linear transformation it induces by right multiplication. If {{mvar|U}} is given the [[discrete topology]], and if {{mvar|U<sup>U</sup>}} is given the [[product topology]], and {{math|End(''<sub>D</sub>U'')}} is viewed as a subspace of {{mvar|U<sup>U</sup>}} and is given the [[subspace topology]], then {{mvar|R}} acts densely on {{mvar|U}} if and only if {{mvar|R}} is [[dense set]] in {{math|End(''<sub>D</sub>U'')}} with this topology.<ref>It turns out this topology is the same as the [[compact-open topology]] in this case. Herstein, p. 41 uses this description.</ref>

==Consequences==
The Jacobson density theorem has various important consequences in the structure theory of rings.<ref>Herstein, p. 41</ref> Notably, the [[Artin–Wedderburn theorem]]'s conclusion about the structure of [[simple ring|simple]] right [[Artinian ring]]s is recovered. The Jacobson density theorem also characterizes right or left [[primitive ring]]s as dense subrings of the ring of {{mvar|D}}-linear transformations on some {{mvar|D}}-vector space {{mvar|U}}, where {{mvar|D}} is a division ring.<ref name="Isaacs187"/>

==Relations to other results==
This result is related to the [[Von Neumann bicommutant theorem]], which states that, for a *-algebra {{mvar|A}} of operators on a [[Hilbert space]] {{mvar|H}}, the double commutant {{mvar|A′′}} can be approximated by {{mvar|A}} on any given finite set of vectors. In other words, the double commutant is the closure of {{mvar|A}} in the [[weak operator topology]]. See also the [[Kaplansky density theorem]] in the von Neumann algebra setting.

==Notes==
{{reflist|2}}

==References==
* {{cite book|author = I.N. Herstein| year = 1968| title = Noncommutative rings| edition =1st| publisher = The Mathematical Association of America| isbn = 0-88385-015-X}}
* {{cite book |last=Isaacs |first=I. Martin |author-link=Martin Isaacs |year=1993 |title=Algebra, a graduate course |edition=1st |publisher=Brooks/Cole |isbn=0-534-19002-2}}
* {{citation|author=Jacobson, N. |title=Structure theory of simple rings without finiteness assumptions |journal=Trans. Amer. Math. Soc. |volume=57 |year=1945 |issue=2 |pages=228–245 |issn=0002-9947 |mr=0011680 |doi=10.1090/s0002-9947-1945-0011680-8|doi-access=free }}

==External links==
*[http://planetmath.org/encyclopedia/JacobsonDensityTheorem.html PlanetMath page]

{{DEFAULTSORT:Jacobson Density Theorem}}
[[Category:Theorems in ring theory]]
[[Category:Module theory]]
[[Category:Articles containing proofs]]