Section 6 Schedule

1.1 The Well-Ordering Principle and Induction

Homework: Read § 1.2, § 1.3; Exercises 1.2, 1.3

1.2 Division with Remainder in \(\mathbb{Z}\)

1.3 Greatest Common Divisors

Homework: Read § 1.4; Exercises 1.4, 1.8, 1.10

1.4 The Fundamental Theorem of Arithmetic

Homework: Read § 2.1, § 2.2 (and Appendix A.8 Relations; Equivalence Relations and Quotient Sets, if necessary); Exercises 1.17, 1.21, 1.23, 1.25

2.1 Equivalence Relations and Quotients

2.2 Congruence mod \(n\)

Homework: Read § 2.3, § 2.4; Exercises 2.1, 2.2, 2.3

2.3 Algebra in \(\mathbb{Z}/n\mathbb{Z}\)

2.4 Properties of the Operations \(+\) and \(\cdot\)

Homework: Read § 3.1; Exercises 2.4, 2.9, 2.10, 2.11

3.1 Definition and Examples

Homework: Read § 3.2, § 3.3; Exercises 3.5, 3.6, 3.7, 3.8

3.2 Basic Properties

3.3 Special Types of Rings

Homework: Read § 4.1, § 4.2; Exercises 3.12, 3.13, 3.14, 3.17, 3.20

4.1 Cartesian Products

4.2 Subrings

Homework: Read § 4.3, § 4.4; Exercises 4.2, 4.3, 4.4, 4.5

4.3 Ring Homomorphisms

4.4 Isomorphisms of Rings

Homework: Read § 5.1, § 5.2; Exercises 4.13, 4.14, 4.19

5.1 Canonical Decomposition, I

5.2 Kernels and Ideals

Homework: Read § 5.3; Exercises 5.4, 5.5, 5.7, 5.9, 5.10, 5.12, 5.13, 5.14, 5.15, 5.16, 5.18

5.3 Quotient Rings

Homework: Read § 5.4, § 5.5; Exercise 5.20

5.4 Canonical Decomposition, II

5.5 The First Isomorphism Theorem

Homework: Read § 5.6, § 5.7; Exercises 5.8, 5.23, 5.25, 5.26

5.6 The Chinese Remainder Theorem

5.7 The Third Isomorphism Theorem

Homework: Exercises 5.33, 5.34, 5.36

Review for the Midterm Exam

Midterm Exam (Chapters 1–5).

Homework: Read § 8.1

8.1 Vector Spaces and Ideals, Revisited

Homework: Read § 8.2; Exercises 8.1, 8.2, 8.3

8.2 The Category of \(R\)-Modules

Homework: Read § 8.3; Exercises 8.6, 8.10, 8.12

8.3 Submodules, Direct Sums

Homework: Read § 8.4; Exercises 8.16, 8.17, 8.19

8.4 Canonical Decomposition and Quotients

Homework: Read § 8.5; Exercises 8.15, 8.23, and prove Claim 8.38

8.5 Isomorphism Theorems

Homework: Read § 10.1; Exercise 8.22

10.1 The Category of Abelian Groups

Homework: Read § 10.2; Exercises 10.1, 10.6

10.2 Cyclic Groups, and Orders of Elements

Homework: Read § 11.1; Exercise 10.7

11.1 Groups and their Category

Homework: Read § 11.2; Exercises 11.1, 11.2, 11.3, 11.4, 11.7

11.2 Why Groups? Actions of a Group

Homework: Read § 11.3; Exercise 11.14

11.3 Cyclic, Dihedral, Symmetric, Free Groups

Homework: Read § 11.4; Exercises 11.28, 11.30, 11.33

11.4 Canonical Decomposition, Normality, and Quotients

Homework: Read § 11.5; Exercises 11.34, 11.36, 11.39

11.5 Isomorphism Theorems

Homework: Exercise 11.40

TBA

Review for the Final Exam

Final Exam (cumulative).