FALL 2019 Math 461 --- RINGS AND
MODULES
Prerequisite: Math
367 or consent of instructor.
Credits: (3-0) 3.
Instructor: Semra Öztürk, M
138, Schedule and office hours are at the address http:/www.metu.edu.tr/~sozkap/aa.pdf.
Catalog Contents: Rings,
ideals, isomorphism theorems, group rings, localization, factor rings. Modules,
submodules, direct products, factor modules. Homomorphisms, classical
isomorphism theorems. The endomorphism ring of a module. Free modules, free
groups. Tensor product of modules. Finitely generated modules over a principal
ideal domain.
Grading will be based on two midterms %40 each,
one final exam %50, and attendence/class
participation %10.
You should take
both
of the midterm exams, I will omit the lower grade
(even if you are happy with your first midterm grade you still have to take the
second midterm exam)
Attendence is
required, attendance and class participation will be %10 of the course grade.
Description of the course: This course is to
provide the background for students who
are willing to learn more about rings which are
the fundamental mathematical structures occuring everywhere ! It is good for everyone J but especially for
students who are planning to study
any algebra related topics such
as algebraic topology, algebraic geometry, even analysis . As the title suggests this course consists of
two parts, rings and modules.
This semester I will spend less than the first half of the
semester on rings and spend more time on modules. We will skip
tensor products completely.
Rings will be a more detailed but
faster version of some of the topics you
have seen in Math 367, and Math
116. Modules will be new to you. They
are generalizations of vector spaces
also generalization of abelian groups.
Thus in module theory linear algebra comes up quite often.
You should be comfortable using linear algebra to get more out of this
course. We will see the primary decomposition theorem for finitely generated modules over a Euclidean
domain.
Modules over group algebras are
examples of groups acting on vector spaces.
About 5-6 weeks we will cover
Rings : rings, subrings, ideals,
quotient rings, annihilators, homorphism of rings, isomorphism theorem for
rings, rings of fractions, Chineese remainder theorem, Euclidean domains,
principal ideal domains, group
algebras, polynomial rings.
About 7-8 weeks we will cover
Modules: modules , submodules, simple modules, Schur’s Lemma, generation
of modules, direct sums, annilators of
modules, free modules, torsion modules, torsion-free modules, finitely generated torsin free modules,
modules over a Euclidean domain or over
a principal ideal domain, primary decomposition theorem, elementary divisors,
invariant factors, (the rational canonical form , if there is enough time and
interest). You can print a tentative syllabus from here.
I will use
two textbooks this semester :
1) Abstract Algebra by David S. Dummit
and Richard M. Foote, Third Edition, only the
Preliminaries and Chapter
7 .
2) A First Course in Module Theory by
M. E. Keating, Imperial Clollege Press, only
the Chapters 2--9
Another book that I used for this
course in previous semesters (you may find it useful as well)
: Introduction to Rings and Modules, Second Revised Edition, by C. Musili, Narosa
Publishing House, 1994, only Chapters 1--5 ( the part for
rings is too long and the part for modules is too short)
A nice set of
lecture notes by Mike Prest which covers more topics than our course content advanced
students may like it,.(Chapters 0,1,2,6 are usefull
for this course, at first reading omit Chapters
3, 4, 5)
Supplementary solved exercises