FALL
2025 Math 461 --- RINGS AND MODULES
Prerequisite:
Math 367 ,
Math 262 or 2360262, 2360267,
2360268 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 and modules which are the fundamental mathematical
structures occuring everywhere ! It is
useful for everyone but especially for students who are planning to study any algebra related topics such as algebraic
topology, algebraic geometry, analysis.
Approximately half of the semester will be on
rings, the second half will be on modules. Rings will be a more
detailed but much 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. (Modules over group algebras rings are examples of groups
acting on vector spaces.) 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. Usually there is not enough time to cover tensor products.
I
will use two textbooks this
semester :
Textbooks : (for the Ring part) Introduction to Rings and
Modules, Second
Revised Edition, by C. Musili, Narosa
Publishing House, 1994 (Chapters 1--5)
(for the module part) : Module
Theory An approach to linear algebra Electronic Edition T.S.Blyth
Some other textbooks/notes which
can be useful:
1) Abstract Algebra by
David S. Dummit and Richard M. Foote,
Third Edition (used as the textbook in an earlier semester, it is better
for graduate courses)
2) A First Course in Module Theory by M.
E. Keating, Imperial Clollege Press (used as the
textbook in an earlier semester)
3) Rings Modules and Linear Algebra Notes by Neil
P. Strickland (unfortunately all rings are commutative
otherwise it is very good)
4) MATH42041/62041: NONCOMMUTATIVEALGEBRA
UNIVERSITYOFMANCHESTER,AUTUMN2019 NOTES BY TOBY STAFFORD,MODIFIED BY MIKE PREST (
Chapters 0,1,2,6)
Supplementary solved exercises