FALL 2018 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/homeworks %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)
There will be only one make -up
exam for any of the exams missed. It will be after the final exam and you
should have your instructor’s permission to be able to take it.
Attendence is
required, attendance and weekly homeworks 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 common mathematical structures occuring everywhere J! 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
the fisrt half of the semester on rings and the second half on
modules. We will skip tensor products completely. Rings will be a more detailed 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. In spirit
module theory is a generalization of linear algebra.
Modules over group algebras are
examples of groups acting on vector spaces.
About 7 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 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 principal ideal domain, primary decomposition theorem,
elementary divisors, invariant factors, (the rational canonical form , if there
is enough time and interest).
We will use one of the
following two books
I have used the two
books below in different semesters, for
this semester I have not decided yet which one to use, please have a look at
both and let me know which one you prefer
for which reason.
Abstract Algebra, Third Edition, by D.S. Dummit and R.M. Foote,
(only Chapters, 0, 7, 8, 9, 10.1, 10,2, 10,3, 12,1, (12.2 if time permits) ). (Syllabus )
Introduction to Rings and
Modules, Second Revised Edition, by C. Musili, Narosa Publishing House, 1994 (Chapters 1—4 are on rings, only Chapter 5 is on modules so I will expand
it a little bit.)
Supplementary
solved exercises