@PROFDOCDIGITALLIBRARIES
ALGEBRA NOTES
Ralph Freese and William DeMeo
March 10, 2015
Contents
I Fall 2010: Universal Algebra & Group Theory 4
1 Universal Algebra 5
1.1 Basic concepts ............................................................................................................................... 5
1.2 Subalgebras and Homomorphisms .............................................................................................. 5
1.3 Direct Products ............................................................................................................................. 6
1.4 Relations ......................................................................................................................................... 7
1.5 Congruence Relations ................................................................................................................... 8
1.6 Quotient Algebras ......................................................................................................................... 9
1.7 Direct Products of Algebras ........................................................................................................ 9
1.8 Lattices ......................................................................................................................................... 10
II Rings, Modules and Linear Algebra 12
2 Rings 13
2.1 The ring Mn(R) ........................................................................................................................... 13
2.2 Factorization in Rings................................................................................................................. 15
2.3 Rings of Frations ......................................................................................................................... 17
2.4 Euclidean Domain and the Eucidean Algorithm ..................................................................... 17
2.5 Polynomial Rings, Gauss’ Lemma ............................................................................................ 17
2.6 Irreducibility Tests ..................................................................................................................... 20
3 Modules 21
3.1 Basics ............................................................................................................................................ 21
3.2 Finitely Generated Modules over a PID ................................................................................... 23
3.3 Tensor Products........................................................................................................................... 34
3.3.1 Algebraic Integers .......................................................................................................... 36
3.4 Projective, Injective and Flat Modules; Exact Sequences..................................................... 37
III Fields 41
1
,@PROFDOCDIGITALLIBRARIES
4 Basics 42
A Prerequisites 43
A.1 Relations ....................................................................................................................................... 43
A.2 Functions ...................................................................................................................................... 43
2
,CONTENTS CONTENTS
Primary Textbook: Jacobson, Basic Algebra [4].
Supplementary Textbooks: Hungerford, Algebra [3]; Dummitt and Foote. Abstract Algebra [1];
Primary Subject: Classical algebra systems: groups, rings, fields, modules (including vector
spaces). Also a little universal algebra and lattice theory.
List of Notation
• AA, the set of maps from a set A into itself.
• Aut(A), the group of automorphisms of an algebra A.
• End(A), the set of endomorphisms an algebra A.
• Hom(A, B), the set of homomorphism from an algebra A into an algebra B.
• Con(A), the set of congruence relations of an algebra A.
• ConA, the lattice of congruence relations of an algebra A.
• Eq(A), the set of equivalence relations of a set A.
• EqA, the lattice of equivalence relations of a set A.
• Sub(A), the set of subalgebras of an algebra A.
• SubA, the lattice of subalgebras of an algebra A.
• SgA(X), the subuniverse generated by a set X ⊆ A.
• N = {1, 2, . . . }, the set of natural numbers.
• Z = {. . . , —1, 0, 1, . . . }, the ring of integers.
• R = (—∞, ∞), the real number field.
• C, the complex number field.
• Q, the rational number field.
3
, Part I
Fall 2010: Universal Algebra & Group Theory
4
ALGEBRA NOTES
Ralph Freese and William DeMeo
March 10, 2015
Contents
I Fall 2010: Universal Algebra & Group Theory 4
1 Universal Algebra 5
1.1 Basic concepts ............................................................................................................................... 5
1.2 Subalgebras and Homomorphisms .............................................................................................. 5
1.3 Direct Products ............................................................................................................................. 6
1.4 Relations ......................................................................................................................................... 7
1.5 Congruence Relations ................................................................................................................... 8
1.6 Quotient Algebras ......................................................................................................................... 9
1.7 Direct Products of Algebras ........................................................................................................ 9
1.8 Lattices ......................................................................................................................................... 10
II Rings, Modules and Linear Algebra 12
2 Rings 13
2.1 The ring Mn(R) ........................................................................................................................... 13
2.2 Factorization in Rings................................................................................................................. 15
2.3 Rings of Frations ......................................................................................................................... 17
2.4 Euclidean Domain and the Eucidean Algorithm ..................................................................... 17
2.5 Polynomial Rings, Gauss’ Lemma ............................................................................................ 17
2.6 Irreducibility Tests ..................................................................................................................... 20
3 Modules 21
3.1 Basics ............................................................................................................................................ 21
3.2 Finitely Generated Modules over a PID ................................................................................... 23
3.3 Tensor Products........................................................................................................................... 34
3.3.1 Algebraic Integers .......................................................................................................... 36
3.4 Projective, Injective and Flat Modules; Exact Sequences..................................................... 37
III Fields 41
1
,@PROFDOCDIGITALLIBRARIES
4 Basics 42
A Prerequisites 43
A.1 Relations ....................................................................................................................................... 43
A.2 Functions ...................................................................................................................................... 43
2
,CONTENTS CONTENTS
Primary Textbook: Jacobson, Basic Algebra [4].
Supplementary Textbooks: Hungerford, Algebra [3]; Dummitt and Foote. Abstract Algebra [1];
Primary Subject: Classical algebra systems: groups, rings, fields, modules (including vector
spaces). Also a little universal algebra and lattice theory.
List of Notation
• AA, the set of maps from a set A into itself.
• Aut(A), the group of automorphisms of an algebra A.
• End(A), the set of endomorphisms an algebra A.
• Hom(A, B), the set of homomorphism from an algebra A into an algebra B.
• Con(A), the set of congruence relations of an algebra A.
• ConA, the lattice of congruence relations of an algebra A.
• Eq(A), the set of equivalence relations of a set A.
• EqA, the lattice of equivalence relations of a set A.
• Sub(A), the set of subalgebras of an algebra A.
• SubA, the lattice of subalgebras of an algebra A.
• SgA(X), the subuniverse generated by a set X ⊆ A.
• N = {1, 2, . . . }, the set of natural numbers.
• Z = {. . . , —1, 0, 1, . . . }, the ring of integers.
• R = (—∞, ∞), the real number field.
• C, the complex number field.
• Q, the rational number field.
3
, Part I
Fall 2010: Universal Algebra & Group Theory
4
