FALL
2026-2027 Math 461 --- RINGS AND MODULES
Prerequisite:
Set 1: 2360262 , 2360367
Set 2: 2360262 , 2360267 , 2360268
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 algebraic topology, algebraic
geometry, 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 the next half 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
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 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). The link for the
ODTU-Syllabus program is here.
Textbook:
: 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 I will do some
omissions from the rings part and and addtitions to the module part from (1) below )
Supplementary books/notes :
1) (including solutions to exercises) RINGS, MODULES AND LINEAR ALGEBRA by N. P. STRICKLAND
2) MODULES OVER PRINCIPAL IDEAL DOMAINS by N.
P. STRICKLAND
3)
Abstract Algebra by David S.
Dummit and Richard M. Foote, Third Edition, only the Preliminaries and Chapter
7 .
4) A First Course in Module Theory by M.
E. Keating, Imperial Clollege Press, only
the Chapters 2--9
5) (additional info related to Chapter 3 of the
textbook) The number of homomorphisms from Zn to Zm.pdf
6) (interesting reading) Classification_of_Finite-Finite rings.pdf
5) (more
advanced material) 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)
6) (may be useful) Supplementary solved exercises