Free Modules - Revision history

Kfagerstrom at 21:38, 9 March 2023

2023-03-09T21:38:30Z

Kfagerstrom at 21:33, 9 March 2023

2023-03-09T21:33:43Z

Kfagerstrom at 17:18, 8 March 2023

2023-03-08T17:18:20Z

Kfagerstrom: /* Theorems: */

2023-03-08T17:15:30Z

Theorems:

Kfagerstrom: /* Theorems: */

2023-03-08T17:12:46Z

Theorems:

Kfagerstrom at 16:56, 8 March 2023

2023-03-08T16:56:11Z

Kfagerstrom at 16:54, 8 March 2023

2023-03-08T16:54:57Z

Kfagerstrom at 16:46, 8 March 2023

2023-03-08T16:46:07Z

Kfagerstrom: Created page with " '''Definition:''' A subset A of an R-module M is a basis of M if A is linearly independent and generates M. An R-module M is a '''free''' R-module if M has a basis."

2023-03-08T05:54:30Z

Created page with " '''Definition:''' A subset A of an R-module M is a basis of M if A is linearly independent and generates M. An R-module M is a '''free''' R-module if M has a basis."

New page

'''Definition:''' A subset A of an R-module M is a basis of M if A is linearly independent
and generates M. An R-module M is a '''free''' R-module if M has a basis.

@@ Line 20: / Line 20: @@
 of R-modules M ∼= N.
-Theorem:(Uniqueness of rank over commutative rings). Let R be a commutative ring
+[[Uniqueness of Rank of Free Modules Over Commutative Rings|Theorem:]] (Uniqueness of rank over commutative rings) Let R be a commutative ring
 with 1 6= 0 and let M be a free R-module. If A and B are both bases for M, then A and B
 have the same cardinality, meaning that there exists a bijection A → B.
@@ Line 37: / Line 37: @@
 R-module.
-Theorem. Every R-module is a quotient of a free $R$-module
+Theorem. Every R-module is a quotient of a free <math>R</math>-module
 [[Category:Modules]] [[Category:Free Modules]]

@@ Line 37: / Line 37: @@
 R-module.
-Theorem. Every R-module is a quotient of a free R-module
+Theorem. Every R-module is a quotient of a free $R$-module
 [[Category:Modules]] [[Category:Free Modules]]

@@ Line 14: / Line 14: @@
 ===Theorems:===
-[[UMP for Free Modules| Theorem:]] (UMP for free modules) Let R be a ring, M be a free R-module with basis B, N be any R-module,
+[[UMP for Free Modules| Theorem:]] (UMP for free modules) Let <math>R</math> be a ring, <math>M</math> a free <math>R</math>-module with basis <math>B</math>, <math>N</math> an <math>R</math>-module, <math>j:B\to N</math> a function. Then there exists a unique <math>R</math>-module homomorphism <math>h:M\to N</math> such that <math>h(b)=j(b)\ \forall b\in B</math>.
 Corollary: If A and B are sets of the same cardinality, and fix a bijection j : A → B.

@@ Line 14: / Line 14: @@
 ===Theorems:===
-Theorem: (UMP for free modules) Let R be a ring, M be a free R-module with basis B, N be any R-module,
+[[UMP for Free Modules| Theorem:]] (UMP for free modules) 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.

@@ Line 14: / Line 14: @@
 ===Theorems:===
-Theorem: Let R be a ring, M be a free R-module with basis B, N be any R-module,
+Theorem: (UMP for free modules) 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.
@@ Line 22: / Line 22: @@
 of R-modules M ∼= N.
 [[Category:Modules]] [[Category:Free Modules]]

← Older revision		Revision as of 16:56, 8 March 2023
Line 1:		Line 1:
−	===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>.	+
		+	===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:===		===Examples:===

@@ Line 1: / Line 1: @@
-'''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>.
+===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:'''
+===Examples:===
 # <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>
@@ Line 10: / Line 10: @@
 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>.

← Older revision		Revision as of 16:46, 8 March 2023
Line 1:		Line 1:
		+	'''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>.

−	'''~~Definition~~:''' ~~A subset A of an~~ R-module M is a basis ~~of M if A is linearly independent~~	+	'''Examples:'''
−	~~and generates M~~. An R-module ~~M is a '''~~free~~'''~~ R-~~module if~~ M ~~has a~~ basis.	+	# <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[x]</math> is a free <math>R</math>-module with (infinite) basis <math>\{1,x,x^2,...,x^i,...\}</math>
		+	# <math>R[x,y]</math> has basis <math>\{x^n,y^m\ \|\ n,m\geq 0 \}</math> as free <math>R</math>-module
		+	# <math>R[x,y]</math> has basis <math>\{1,y,y^2, ...,y^i,...\}</math> as free <math>R[x]</math>-module
		+
		+
		+	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>.