MAT3207: Rings and Modules

This course is a rejoinder to the course MAT2203 Abstract Algebra, and has some Number Theoretic ideas in some areas. You do not need Number Theory for this course. The latter course deals with properties of integers without use of techniques from other mathematical fields (Elementary Number Theory). This course centers on algebraic number theory in which numbers are roots of polynomials with rational coefficients. This course includes the following topics: Ring; Congruences; Integral domains and Fields; Factorization; and Rings of Polynomials. 
 
Course Objectives
On successful completion of this course unit, the learners should be able: 

• Understand the connection between primes and irreducible in an arbitrary integral `s domain 
• Investigate whether an integral domain is a unique factorization domain. When it is not, to be able to find essentially different factorizations of a given element and to prove the factorizations essentially different 
• For an integral domain which is not a unique factorization domain, to be able to find essentially different factorizations of a given element and to prove the factorizations essentially different 
• Find greatest common divisors and least common multiples and to decide when they are unique (up to associates) 
• Work with rings of polynomials over finite fields 
• Prove that an ideal is prime and to write ideals as products of prime ideals 
• Work with the ring Z[ n] 
• Work with the ring R[X] where R is an integral domain ix. Understand 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 

Expected Learning Outcome
This course unit is meant: 

• To help the student centre on algebraic number theory in which numbers are roots of polynomials with rational coefficients.  
• To provide an introduction to commutative ring theory.  
• To help learners get familiar to concepts, such as factorization, primness, divisibility etc., in a new, more general setting of commutative rings.  
• To help work with quotient rings and extension of fields. In addition, the course includes topics from: rings of quotients, finite fields and extensions of fields.  
• To provide instruction that contributes to the learners’ abilities to think critically and solve real life problems, to reason mathematically and apply computational skills.