Difference between revisions of "Free Modules"
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| − | + | ===Definition:=== An <math>R</math>-module <math>M</math> is a '''free''' <math>R</math>-module if <math>M</math> has a basis. A subset <math>A</math> of an <math>R</math>-module <math>M</math> is a basis of <math>M</math> if <math>A</math> is linearly independent and generates <math>M</math>. | |
| − | + | ===Examples:=== | |
# <math>R=R\{1_R\}</math> is a free <math>R</math>-module | # <math>R=R\{1_R\}</math> is a free <math>R</math>-module | ||
# <math>R^2=R\oplus R</math> has basis <math>\{(1_R,0_R),(0_R,1_R)\}</math> | # <math>R^2=R\oplus R</math> has basis <math>\{(1_R,0_R),(0_R,1_R)\}</math> | ||
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All free <math>R</math>-modules <math>M </math> have some basis <math>B=\{b_1,...,b_n\} </math> so have rank <math>n </math>, and can be written as <math>R^n\cong M </math>. Additionally every element <math>m\in M </math> can be uniquely written <math> m=\sum_{i=1}^n r_ib_i</math>. | All free <math>R</math>-modules <math>M </math> have some basis <math>B=\{b_1,...,b_n\} </math> so have rank <math>n </math>, and can be written as <math>R^n\cong M </math>. Additionally every element <math>m\in M </math> can be uniquely written <math> m=\sum_{i=1}^n r_ib_i</math>. | ||
| + | |||
| + | ===Theorems:=== | ||
| + | Theorem: Let R be a ring, M be a free R-module with basis B, N be any R-module, | ||
| + | and let j : B → N be any function. Then there is a unique R-module homomorphism | ||
| + | h : M → N such that h(b) = j(b) for all b ∈ B. | ||
| + | |||
| + | Corollary: If A and B are sets of the same cardinality, and fix a bijection j : A → B. | ||
| + | If M and N are free R-modules with bases A and B respectively, then there is an isomorphism | ||
| + | of R-modules M ∼= N. | ||
| + | |||
| + | |||
| + | |||
| + | [[Category:Modules]] [[Category:Free Modules]] | ||
Revision as of 16:54, 8 March 2023
===Definition:=== An \(R\)-module \(M\) is a free \(R\)-module if \(M\) has a basis. A subset \(A\) of an \(R\)-module \(M\) is a basis of \(M\) if \(A\) is linearly independent and generates \(M\).
Examples:
- \(R=R\{1_R\}\) is a free \(R\)-module
- \(R^2=R\oplus R\) has basis \(\{(1_R,0_R),(0_R,1_R)\}\)
- \(R[x]\) is a free \(R\)-module with (infinite) basis \(\{1,x,x^2,...,x^i,...\}\)
- \(R[x,y]\) has basis \(\{x^n,y^m\ |\ n,m\geq 0 \}\) as free \(R\)-module
- \(R[x,y]\) has basis \(\{1,y,y^2, ...,y^i,...\}\) as free \(R[x]\)-module
All free \(R\)-modules \(M \) have some basis \(B=\{b_1,...,b_n\} \) so have rank \(n \), and can be written as \(R^n\cong M \). Additionally every element \(m\in M \) can be uniquely written \( m=\sum_{i=1}^n r_ib_i\).
Theorems:
Theorem: Let R be a ring, M be a free R-module with basis B, N be any R-module, and let j : B → N be any function. Then there is a unique R-module homomorphism h : M → N such that h(b) = j(b) for all b ∈ B.
Corollary: If A and B are sets of the same cardinality, and fix a bijection j : A → B. If M and N are free R-modules with bases A and B respectively, then there is an isomorphism of R-modules M ∼= N.
