Section 2 Course Information

Subsection 2.1 Course Description

This course will utilize the logical foundation and proof techniques developed in prerequisite courses to explore the basic constructions in modern algebra: rings, modules, and groups. Following our textbook, we treat rings first, since their arithmetic is closest to that of the familiar integers, and take up groups later in the term. The primary goal of this course is to gain familiarity with these basic algebraic structures and how they interact with one another (actions, homomorphisms, etc.). An equally important goal of this course is to improve students’ ability to read and write mathematical proofs.

Subsection 2.2 Course Prerequisites

A grade of C or better in MATH 3073 (Previously MATH 307) or MATH 3113 (Previously MATH 311) is required to enroll in this course.

In particular, students are expected to have experience with the following topics.

propositional logic, such as
- conjunction,
- disjunction,
- negation,
- conditionals, biconditionals and logical equivalence,
- contraposition
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first-order logic, such as
- predicates,
- universal quantification,
- existential quantification
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basic set theory, such as
- (indexed) union,
- (indexed) intersection,
- complement,
- functions
  - domain and codomain,
  - composition,
  - injectivity,
  - surjectivity,
  - bijectivity
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- equivalence relations and classes
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basic proof techniques, such as
- direct proof,
- proof by contraposition,
- proof by contradiction,
- proof by induction
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Subsection 2.3 Student Learning Outcomes

Upon successful completion of this course, students will be able to

state the definitions and fundamental theorems concerning rings, modules, and groups,
verify whether a given set with operations forms a ring, module, or group, and whether a given map is a homomorphism,
compute in concrete examples, including modular arithmetic, polynomial rings, and cyclic groups,
construct examples and counterexamples distinguishing algebraic structures and their properties,
write clear, logically correct proofs of statements in elementary abstract algebra, and identify gaps or errors in a proposed proof.

Subsection 2.4 Course Topics

This course can be divided into three parts; a week-by-week plan is available in the Schedule.

Rings: We first explore the basic properties and arithmetic of the integers, including modular arithmetic. We generalize these properties to other sets equipped with operations that “behave like” addition and multiplication, then study “structure preserving” functions and constructions with rings. Depending on time and interest, we may also explore some classes of “nice” rings (e.g. integral domains, unique factorization domains, principal ideal domains, polynomial rings).
Modules: We generalize the familiar notion of a vector space from linear algebra by allowing our scalars to come from a ring rather than a field. We study the analogous “structure preserving” functions and constructions for these objects.
Groups: We consider the structures that arise from a set equipped with a single operation. As with rings and modules, we study the analogous “structure preserving” functions and constructions.

Subsection 2.5 Instructional Methods

This course is offered as a face-to-face course. Learning will be facilitated through traditional lecture, group work/activities, homework, and in-class assessments.

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