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  

 The number of homomorphisms from Zn to Zm.pdf 

 Classification_of_Finite  rings.pdf