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

Code	Subject	
MML1005	Special Topics of Modern Algebra	

Section	Semester	Hours: C+S+L	Category	Type	
Mathematics	4	2+2+0	speciality	optional	

Teaching Staff in Charge	Lect. MODOI Gheorghe Ciprian, Ph.D., cmodoi@math.ubbcluj.ro	
Aims	The course should teach the students the notions of Morita equivalence and duality, tilting and cotilting module. The students must be also able to apply creatively the learned theory, in order to establish equivalences or dualities between particular classes of modules.  	
Content	1. Preliminaries on module theory. Equivalences and dualities of categories.
2. Rings of matrices. Equivalences of module categories. Generators and Morita contexts. 
Morita theorem. 
3. Cogenerators. Pontryagin duality. Morita duality. 
4. Classes of modules closed under different constructions (epimorphisms, direct sums etc.).
(Pre)torsion classes. Tilting modules. Tilting theorem. Cotorsion classes. Cotilting modules. Cotilting theorem.  	
Bibliography	1. F. Anderson, K. Fuller, Rings and Categories of Modules, Springer Verlag, 1973.
2. R.R. Colby, K.R. Fuller, Equivalence and Duality for Module Categories, Cambridge Univ. Press, Cambridge, 2004.
3. R. Goebel, J. Trlifaj, Approximations and Endomorphism Algebras of Modules, W. de Gruyter, 2006.
4. T.Y. Lam, Lectures on Modules and Rings, Springer-Verlag, New York, 1999.
5. T.Y. Lam, Exercices in Modules and Rings, Problem Books in Mathematics, Springer-Verlag, 2007.
6. B. Stenstrom, Rings of Quotients, Springer-Verlag, Berlin, 1975.
7. J. Trlifaj, Covers, envelopes and cotorsion theories, Lecture notes for the workshop $Homological Methods in Module Theory$, Cortona, 2000.
	
Assessment	The students will get the grade as follows: The students will get points (from 0.5 to 5) for homeworks (exercices given during the course). The points for an exercise will be awarded to only one student. 10 points are equal to the grade 10 and so on. Additionally there is a writen exam for those students which are not satisfied by the grade obtained by homeworks. The subjects of the exam consists from basic definitions from the course and some exercises chosen from the homeworks.