Subject: Special Topics in Module Theory	Code: MA028	

Semester	 Hours: C+S+L	Type	 Section	
7	2+2+0	optional	Mathematics	
7	2+2+0	optional	Mathematics-Computer Science	

Teaching Staff in Charge	Assoc.Prof. BREAZ Simion Sorin, Ph.D., bodo@math.ubbcluj.ro	
Aims	We shall present basic notions in Module Theory with applications in Abelian Group Theory and in Basic Linear Algebra. The students will solve problems concerning the general theory and they will apply these results to the mentioned particular cases	
Content	1. Notions of ring theory.
2. Modules, basic properties. Submodules and factor modules. Homomorphisms, monomorphisms, epimorphisms. Exact sequences.
3. Direct products and direct sums of modules. Direct summands. Essential submodules.
4. Free modules. Universal property of free modules. The construction of a module using generators and relations.
5. Artinian and noetherian modules. Finitely generated modules.
6. Simple and semisimple modules. The socle and the radical.
7. Projective modules. Characterizations. Every projective module ovel a local ring is free. The abelian groups case.
8. Injective modules. Caracterizations. Self-injective modules. The abelian groups case.
9. Tensorial product. Flat modules. The abelian groups case	
Selective References	1. F. KASCH, A. MADER: Rings, modules, and the total. Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2004. 
2. R. COLBY, K. FULLER: Equivalence and duality for module categories. With tilting and cotilting for rings. Cambridge Tracts in Mathematics, 161. Cambridge University Press, Cambridge.
3. A. FACCHINI: Module theory. Endomorphism rings and direct sum decompositions in some classes of modules. Progress in Mathematics, 167. Birkhäuser Verlag, Basel, 1998. 
4.  I. PURDEA: Tratat de algebra moderna. Vol. II. Editura Academiei Republicii Socialiste România, Bucharest, 1982. 
5. G. CALUGAREANU, S. BREAZ, C. MODOI, C. PELEA, D. VALCAN: Exercises in abelian group theory. Kluwer Texts in the Mathematical Sciences, 25. Kluwer Academic Publishers Group, Dordrecht, 2003.  
6. F. ANDERSON, K. FULLER: Rings and categories of modules. Second edition. Graduate Texts in Mathematics, 13. Springer-Verlag, New York, 1992. 
7. E. ENOCHS, R. LOPEZ:  Gorenstein flat modules. Nova Science Publishers, Inc., Huntington, NY, 2001.  
8. C. NASTASESCU: Teoria dimensiunii în algebra necomutativa. Editura Academiei Republicii Socialiste România, Bucharest, 1983.   
9. C. NASTASESCU: Inele. Module. Categorii. Editura Academiei Republicii Socialiste România, Bucharest, 1976. 
10. G. CALUGAREANU: Lattice concepts of module theory. Kluwer Texts in the Mathematical Sciences, 22. Kluwer Academic Publishers, Dordrecht, 2000.	
Assessment	Test in the IX-th week (20%x final grade). Exam (80%x final grade)