Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT

Code	Subject	
MML1002	Module Theory	

Section	Sem.	Hours: C+S+L	Category	Type	
Mathematics	2	2+1+0	speciality	compulsory	

Teaching Staff in Charge	Assoc.Prof. BREAZ Simion Sorin, Ph.D., bodo@math.ubbcluj.ro	
Aims	We will present basic facts about modules over associative rings, principal ideal domains. We will also present recent results from the general Module Theory in order that the students will be able to use these results to understand special topics in algebra. By
the graduation of this class, the students will get the following competences:
 - Understanding basic notions as direct sum, direct product,
 tensorial product;
 - They will be able to use the injective hull and the projective
 cover;
 - They will construct and use injective (projective) resolutions;
 - They will use special classes of submodules (supramodules) in
 the study of modules.
	
Content	 1. Basic notions
 2. Direct sums
 3. Direct products
 4. Free and projective modules
 5. Injective modules
 6. Semisimple rings and modules
 7. Finiteness conditions
 8. Noetherian/artinian rings and modules
 9. Tensorial product
10. Flat modules. Pure submodules
11. Modules over PID
12. Rings and modules of quotients
	
Bibliography	1.Anderson, F.W., Fuller, K.R.: Rings and Categories of Modules,
Graduate Texts in Math. Vol. 13, Springer-Verlag, 1992.

2.Lam, T.Y.: Lectures On Modules and Rings, Graduate Texts in
Math. Vol. 189, Springer-Verlag, 1999.

3.Lam, T.Y.: A First Course in Noncommutative rings, Graduate
Texts in Math. Vol. 131, Springer-Verlag, 1991.

4.Lam, T.Y.: Exercices in Classical Ring Theory, Problem Books in
Mathematics, Springer-Verlag, 1995.

5.Lam, T.Y.: Exercices in Modules and Rings, Problem Books in
Mathematics, Springer-Verlag, 2007.

6.Stenstrom, B.: Ring of Quotients, Graduate Texts in Math.,
Springer-Verlag, 1975.

7.Wickless, W.: A First Course in Graduate Algebra, Taylor and
Francis, 2004.
	
Assessment	A written final exam (grade E), a test at the seminar (grade T)
and a referee (grade R). The exam subjects have theoretical
questions from all the studied topics, and one problem, among the
problems studied at the course and last 4 seminars. The test
subject have practical questions (exercices and problems) from
topics studied in first 10 weeks. The final grade is the weighted
mean of the three grades mentioned above, conditioned by all the
grades being at least 5 from 10. Otherwise, the exam will not be
passed.
The final grade = 50%E + 25%L + 25%R.