Jacobson's tightness theorem

The leak rate of Jacobson , named after Nathan Jacobson , is a mathematical theorem from the representation theory with applications in ring theory and group theory . It was first proven by Jacobson in 1945 and establishes a close relationship between certain rings and die rings over oblique bodies .

Definitions

Let it be a ring with a single element and a left- R module . Such a module is called simple if it does not contain any non-trivial sub- modules, i.e. in addition to and no further sub- modules . The module is called loyal if only applies to. ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle \{0\}}$ ${\displaystyle M}$ ${\displaystyle rM=\{0\}}$ ${\displaystyle r=0}$

${\displaystyle \mathrm {End} _{R}(M)}$let the ring of R endomorphisms be on . Then by definition ${\displaystyle M}$ ${\displaystyle M}$

${\displaystyle \alpha \cdot m:=\alpha (m)}$   For   ${\displaystyle \alpha \in \mathrm {End} _{R}(M),m\in M}$

to a module and one can speak of -linear mapping. ${\displaystyle \mathrm {End} _{R}(M)}$ ${\displaystyle \mathrm {End} _{R}(M)}$

The linearity of endomorphisms just means ${\displaystyle R}$ ${\displaystyle \alpha \in \mathrm {End} _{R}(M)}$

${\displaystyle \alpha (rm)=r\alpha (m)}$   for everyone   . ${\displaystyle r\in R,m\in M,\alpha \in \mathrm {End} _{R}(M)}$

Denoting by the links multiplication on , one equation above can also be read so that each a linear map is. Note that this is generally non- linear if it is not commutative. ${\displaystyle \ell _{r}}$ ${\displaystyle r}$ ${\displaystyle M}$ ${\displaystyle \ell _{r}}$ ${\displaystyle \mathrm {End} _{R}(M)}$ ${\displaystyle \ell _{r}}$ ${\displaystyle R}$ ${\displaystyle R}$

Formulation of the sentence

Let it be a ring with one element, a simple, faithful links module and a linear mapping. ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle \varphi }$ ${\displaystyle \mathrm {End} _{R}(M)}$

Then there is finally one with for everyone . ${\displaystyle m_{1},\ldots ,m_{n}\in M}$ ${\displaystyle r\in R}$ ${\displaystyle \varphi (m_{i})=rm_{i}}$ ${\displaystyle i=1,\ldots ,n}$

In words: every linear mapping behaves on a finite set like the left multiplication with a ring element. ${\displaystyle \mathrm {End} _{R}(M)}$

Remarks

Because of the simplicity of the left- module , according to Schur's lemma, it is a skew body . For each and every one ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle D=\mathrm {End} _{R}(M)}$ ${\displaystyle m\in M}$ ${\displaystyle \varphi \in \mathrm {End} _{D}(M)}$

${\displaystyle B(m,\varphi ):=\{\psi \in \mathrm {End} _{D}(M)\mid \varphi (m)=\psi (m)\}}$.

Then forming the sub-base of a topology on which one the finite topology calls. ${\displaystyle B(m,\varphi )}$ ${\displaystyle \mathrm {End} _{D}(M)}$

In the situation of the sentence is and because of the loyalty one can identify with . In this sense, and the above sentence says that it is dense with respect to the finite topology. Hence the name “ Dichtheitssatz” . ${\displaystyle \ell _{r}\in \mathrm {End} _{D}(M)}$ ${\displaystyle r\in R}$ ${\displaystyle \ell _{r}}$ ${\displaystyle R\subset \mathrm {End} _{D}(M)}$ ${\displaystyle R\subset \mathrm {End} _{D}(M)}$

Another peculiarity is when there is a finite-dimensional vector space. If one chooses a vector space basis in the above theorem , then every endomorphism is already uniquely determined by its values ​​on the , and this results from Jacobson's density theorem . ${\displaystyle M}$ ${\displaystyle D}$ ${\displaystyle m_{1},\ldots ,m_{n}\in M}$ ${\displaystyle \mathrm {End} _{D}(M)\cong M_{n}(D)}$ ${\displaystyle m_{i}}$ ${\displaystyle R\cong M_{n}(D)}$

Primitive rings

A ring with a single element is called primitive if it has a faithful, simple module. Jacobson's theorem of density, together with the above remark, states that there is an inclined body and a module for a primitive ring , so that in is close , because the faithful, simple module that exists according to the definition does what is required. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle D}$ ${\displaystyle D}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle \mathrm {End} _{D}(M)}$

This property characterizes primitive rings because, conversely , if a module is tight for a module over an oblique body , then as a module is true, because , and because of the tightness also simple. ${\displaystyle R\subset \mathrm {End} _{D}(M)}$ ${\displaystyle M}$ ${\displaystyle D}$ ${\displaystyle M}$ ${\displaystyle R}$ ${\displaystyle R\subset \mathrm {End} _{D}(M)}$

This characterization of primitive rings is ultimately nothing more than an alternative formulation of Jacobson's theorem of density, the latter can therefore also be found in this form. Jacobson formulated the sentence twice in his textbook Basic Algebra , first as above and then as a characterization of primitive rings under the name Density Theorem for Primitive Rings .

Group representations

As a further application example we show a sentence that goes back to Burnside .

• Let be a group and an n -dimensional, irreducible representation over the field of complex numbers. Then there are such that - are linearly independent . ${\displaystyle G}$ ${\displaystyle \rho :G\rightarrow \mathrm {GL} (\mathbb {C} ^{n})}$ ${\displaystyle g_{1},\ldots ,g_{n^{2}}\in G}$ ${\displaystyle \rho (g_{1}),\ldots ,\rho (g_{n^{2}})}$ ${\displaystyle \mathbb {C} }$

We consider group algebra and the canonical extension from to an algebra homomorphism . Be . These definitions make it a faithful module that is simple because of the assumed irreducibility. It is according to Schur's lemma together with the algebraic closure of . From Jacobson's density theorem and the following remark it now follows that the -dimensional vector space is generated as the vector space by the . The assertion now follows from the basic selection theorem. ${\displaystyle \mathbb {C} [G]}$ ${\displaystyle \rho }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle {\tilde {\rho }}:\mathbb {C} [G]\rightarrow \mathrm {End} _{\mathbb {C} }(\mathbb {C} ^{n})}$ ${\displaystyle R={\tilde {\rho }}(\mathbb {C} [G])\subset \mathrm {End} _{\mathbb {C} }(\mathbb {C} ^{n})}$ ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle R}$ ${\displaystyle D=\mathrm {End} _{R}(M)=\mathbb {C} }$ ${\displaystyle \mathbb {C} }$ ${\displaystyle R=\mathrm {End} _{\mathbb {C} }(\mathbb {C} ^{n})}$ ${\displaystyle n^{2}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \mathrm {End} _{\mathbb {C} }(\mathbb {C} ^{n})}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle \rho (g),g\in G}$

This statement can be used for counting arguments. In the textbook by Derek JS Robinson given below, it is explained how this results in Schur's theorem, according to which every torsion group in is finite. ${\displaystyle \mathrm {GL} (n,\mathbb {Q} )}$

See also

• The Transitivitätssatz of Kadison is an analog tightness statement for irreducible representations of C * -algebras on Hilbert spaces .

Individual evidence

1. ^ N. Jacobson: Structure Theory of Simple Rings Without Finiteness Assumptions , Transactions of the American Mathematical Society, Vol. 57, No. 2 (1945), pp. 228-245
2. ^ Derek JS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 8.1.7: The Jacobson Density Theorem
3. ^ I. Martin Isaacs: Algebra - A Graduate Course , American Mathematical Society, Graduate Studies in Mathematics (2009), Volume 100, Theorem (13.14)
4. ↑ Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), Theorem 2.1.6 with explanation above
5. ↑ Louis H. Rowen: Ring Theory. Volume 1. Academic Press Inc., Boston et al. 1988, ISBN 0-125-99841-4 ( Pure and Applied Mathematics 127), definition 2.1.1.
6. ↑ Benson Farb, R. Keith Dennis: Noncommutative Algebra , Springer-Verlag (1993), ISBN 978-0-387-94057-1 , Theorem 5.2 (Jacobson Density Theorem)
7. ^ N. Jacobson: Basic Algebra II , Dover Publications Inc. (1980), Chapter 4.3: Density Theorems
8. ^ Derek JS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 8.1.8
9. ^ Derek JS Robinson: A Course in the Theory of Groups , Springer-Verlag 1996, ISBN 0-387-94461-3 , sentence 8.1.11