# Rings and Modules

Instructor:
Course Venue:

## Lecture Theatre LLT 2B

CIT Block B
LLT 2B
College of Computing and Information Sciences
Course Code:
MTH 3107
Course Credit Units:
3
Semester:
Semester 1
Year of Study:
Year 3
Course Description & Objectives:

This course is a rejoinder to the course MTH 2201 Abstract Algebra, and in a way related to course MTH 3214 Number Theory in some areas. The latter course deals with properties of integers without use of techniques from other mathematical fields (Elementary Number Theory). This course centres on algebraic number theory in which numbers are roots of polynomials with rational coefficients. This course will provide an introduction to commutative ring theory.  Students will study familiar concepts, such as factorisation, primness, divisibility etc., in a new, more general, setting of commutative rings.  In addition, the course includes topics from: rings of quotients, finite fields and extensions of fields.

Learning Outcomes:

By the end of this course, the student should be able to:

• Write elements of a factorization domain as products of irreducibles
• Understand the connection between primes and irreducibles in an arbitrary integral domain
• Investigate whether an integral domain is a unique factorisation domain. When it is not, to be able to find essentially different factorisations of a given element and to prove the factorisations essentially different
• For an integral domain which is not a unique factorisation domain, to be able to find essentially different factorisations of a given element and to prove the factorisations essentially different
• Find greatest common divisors and least common multiples and to decide when they are unique (up to associates)
• Prove that an ideal is prime and to write ideals as products of prime ideals
• Explain the construction of the ring of quotients of an integral domain and its connection with the construction of the rational numbers
• Demonstrate mastery of the concepts by constructing proofs of simple theorems.
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