Ring Theory (MA416/MA538)

Hilbert

This is the homepage of MA416/MA538 for the academic year 2009-2010. The page will be regularly updated throughout the first semester. Any suggestions for improving it are welcomed by its author, Rachel Quinlan.

Contents

Lecturer
Course Content
Course Activities - what students do
Learning Outcomes
Assessment and Feedback
Syllabus
Course Notes

Lecturer for Ring Theory (MA416/MA538)

Dr Rachel Quinlan
Office : Room 105, Ground Floor, Áras de Brún
Phone : (49)3796
email : rachel.quinlan@nuigalway.ie

Course Content

This course is an introduction to ring theory. The philosophy of this subject is that we focus on similarities in arithmetic structure between sets (of numbers, matrices, functions or polynomials for example) which might look initially quite different but are connected by the property of being equipped with operations of addition and multiplication. The set of integers and the set of 2 by 2 matrices with real numbers as entries are examples of rings. These sets are obviously not the same, but they have some similarities - and some differences - in terms of their algebraic structure. Although people have been studying specific examples of rings for thousands of years, the emergence of ring theory as a branch of mathematics in its own right is a very recent development. Much of the activity that led to the modern formulation of ring theory took place in the first half of the 20th century. Ring theory is powerful in terms of its scope and generality, but it can be simply described as the study of systems in which addition and multiplication are possible.

Course Activities - what students do

There will be three components to the course activities.
  1. Study of the course text
    The course text is the set of lecture notes that is available at this website. Students are expected to engage in independent study of the lecture notes. Not everything in the lecture notes will be discussed in detail in seminars, but everything in the lecture notes is on the course syllabus. Of course this set of notes is not the only (or the best) source for this material; you will probably also wish to consult other texts and/or online information about ring theory.
  2. Seminar Series
    We will have two weekly seminars. Seminars will be held in C219 at 1.00 on Tuesdays and at 4.00 on Wednesdays. The online dictionary dictionary.com defines seminar as follows :
    ``a small group of students, as in a university, engaged in advanced study and original research with a member of the faculty and meeting regularly to exchange information and hold discussions.''
    Each seminar session will focus on a particular section or sections of the lecture notes; these should be studied in advance of the seminar. Topics for seminar discussion will include :
  3. Ring Theory Learning Journal
    Each student will maintain a journal for completing assigned tasks. Five ``journal tasks'' will be assigned as the course proceeds. Each will be concerned with a specific topic, and will involve writing three to five pages of notes including examples, definitions, non-examples, explanations, observations and so on. For each task, some explicit instructions will be given on specific items that should be included, but plenty of scope will be allowed for creativity and personal judgement.

Note on Workload
This module accounts for 5 ECTS credits. One ECTS credit is considered to equate to 25 to 30 hours of work by a student. This means that you should spend 125 to 150 hours in total working on Ring Theory, including time spent attending seminars and the final exam. If (for example) you spend 6 hours per week working on Ring Theory plus two hours at seminars, that will account for 96 hours over the twelve weeks of term.

Learning Outcomes

Upon successful completion of this module students will be able to

Assessment and Feedback

30% of the marks in Ring Theory will be awarded for continuous assessment, consisting of participation in seminars (10% ) and completion of the learning journal (20% ).
The remaining 70% will be awarded for the final examination. Undergraduate students registered for MA416 will have their final exam in the Summer Examination period. Postgraduate students (H.Dip, Year 1 of two-year M.A.) registered for MA538 will have their final exam in the Winter Examination period. More details about the exam format and content will be provided later, but the examination questions will be related to the journal tasks and seminar discussions as well as to the lecture notes. During the first four weeks of term students will have the opportunity to receive formative feedback and advice on the quality of their journal entries. This feedback will be provided in individual meetings with the lecturer. No marks will be awarded (or deducted!) based on work submitted at this stage. The deadline for handing in journals for formative feedback will be Tuesday October 7. Journals will be handed in again after the first four tasks have been completed (approximately Week 8 of term). At this point marks will be assigned, and again feedback will be provided. Journals will be handed in for final assessment at the last seminar session.

Outline Syllabus for MA 416

Chapter 1 What is a Ring?
Chapter 2 Factorization in Polynomial Rings
Chapter 3 Ideals, Homomorphisms and Factor Rings
Chapter 4 Unique Factorization Domains (UFDs)
Chapter 5 Further Topics in Ring Theory

I hope that the lecture notes, which will be posted at this site as the course proceeds, will be sufficient as a "text" for this course. For supplementary reading I suggest the following books, which are available in the library.

Course Notes

Outline notes will be posted here throughout the year as material is covered in class. These notes will contain most of the material discussed in lectures.

Chapter 1

Chapter 2

Chapter 3

Section 2.1

Section 3.1

Section 2.2

Section 3.2

Section 3.3

Section 3.4

Note The picture at the top of this page is of an Congolese stamp issued in 2001, honouring David Hilbert. For more stamps featuring mathematics and mathematicians, see this site.
NUI, Galway Department of Mathematics