Algebra Notes – Universal Algebra, Rings, Modules and Fields, Lecture Notes, Spring 2015

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ALGEBRA NOTES
Ralph Freese and William DeMeo

March 10, 2015


Contents

I Fall 2010: Universal Algebra & Group Theory 4

1 Universal Algebra 5
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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
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1.7 Direct Products of Algebras ........................................................................................................ 9
1.8 Lattices ......................................................................................................................................... 10
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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
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4 Basics 42

A Prerequisites 43
A.1 Relations ....................................................................................................................................... 43
A.2 Functions ...................................................................................................................................... 43



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,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.
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• 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.
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• SubA, the lattice of subalgebras of an algebra A.

• SgA(X), the subuniverse generated by a set X ⊆ A.
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• 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.




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, Part I
Fall 2010: Universal Algebra & Group Theory

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