example of free module with bases of diffrent cardinality
Let k be a field and V be an infinite dimensional vector space over k. Let {ei}i∈I be its basis. Denote by R=End(V) the ring of endomorphisms of V with standard addition
and composition as a multiplication.
Let J be any set such that |J|≤|I|.
Proposition. R and ∏j∈JR are isomorphic as a R-modules.
Proof. Let α:I→J×I be a bijection (it exists since |I|≥|J| and I is infinite) and denote by π1:J×I→J and π2:J×I→I the projections
. Moreover let δ1=π1∘α and δ2=π2∘α.
Recall that ∏j∈JR={f:J→R} (with obvious R-module structure) and define a map ϕ:∏j∈JR→R by defining the endomorphism
ϕ(f)∈R for f∈∏j∈JR as follows:
ϕ(f)(ei)=f(δ1(i))(eδ2(i)). |
We will show that ϕ is an isomorphism. It is easy to see that ϕ is a R-module homomorphism
. Therefore it is enough to show that ϕ is injective
and surjective
.
1) Recall that ϕ is injective if and only if ker(ϕ)=0. So assume that ϕ(f)=0 for f∈∏j∈JR. Note that f=0 if and only if f(j)=0 for all j∈J and this is if and only if f(j)(ei)=0 for all j∈J and i∈I. So take any (j,i)∈J×I. Then (since α is bijective) there exists i0∈I such that α(i0)=(j,i). It follows that δ1(i0)=j and δ2(i0)=i. Thus we have
0=ϕ(f)(ei0)=f(δ1(i0))(eδ2(i0))=f(j)(ei). |
Since j and i were arbitrary, then f=0 which completes this part.
2) We wish to show that ϕ is onto, so take any h∈R. Define f∈∏j∈JR by the following formula:
f(j)(ei)=h(eα-1(j,i)). |
It is easy to see that ϕ(f)=h. □
Corollary. For any two numbers n,m∈ℕ there exists a ring R and a free module M such that M has two bases with cardinality n,m respectively.
Proof. It follows from the proposition, that for R=End(V) we have
Rn≃R≃Rm. |
For finite set J module ∏j∈JR is free with basis consisting |J| elements (product
is the same as direct sum
). Therefore (due to existence of previous isomorphisms) R-module R has two bases, one of cardinality n and second of cardinality m. □
Title | example of free module with bases of diffrent cardinality |
---|---|
Canonical name | ExampleOfFreeModuleWithBasesOfDiffrentCardinality |
Date of creation | 2013-03-22 18:07:18 |
Last modified on | 2013-03-22 18:07:18 |
Owner | joking (16130) |
Last modified by | joking (16130) |
Numerical id | 13 |
Author | joking (16130) |
Entry type | Example |
Classification | msc 16D40 |
Related topic | IBN |