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

 Code Subject MML0004 Algebra 2. (Basic Algebraic Structures)
 Section Semester Hours: C+S+L Category Type Mathematics 2 2+2+0 fundamental compulsory Mathematics and Computer Science 2 2+2+0 fundamental compulsory Applied Mathematics 2 2+2+0 fundamental compulsory
 Teaching Staff in Charge
 Prof. MARCUS Andrei, Ph.D.,  marcusmath.ubbcluj.roAssoc.Prof. BREAZ Simion Sorin, Ph.D.,  bodomath.ubbcluj.roLect. MODOI Gheorghe Ciprian, Ph.D.,  cmodoimath.ubbcluj.ro
 Aims Basic notions and results concerning algebraic structures.
 Content Chapter I. GROUPS 1. Groups, homomorphisms, subgroups: basic results and examples. 2. The symmetric group. 3. Lattice of subgroups. Generated subgroups. 4. Cyclic groups, order of an element, dihedral group. 5. Equivalences induced by a subgroup, Lagrange's theorem. 6. Normal subgroups. Factor group. Examples. 7. Isomorphism theorems. 8. Inner automorphisms, conjugation classes. 9. Products of groups and subgroups. 10. Classification of groups of small order. Chapter II. RINGS AND FIELDS 1. Rings and fields: basic results and examples. 2. Homomophisms, subrings, subfields. 3. Residues mod n. Function rings, matrix rings. 4. Rings of polynomials. Chapter III. MODULES AND ALGEBRAS 1. Modules over commutative rings. Algebras. 2. Homomorphisms, submodules, subalgebras.
 References 1. I. PURDEA, I. POP, Algebra, Editura GIL, Zalau, 2003. 2. I.D. ION, N. RADU, Algebra (ed.4), Ed. Didactica si Pedagogica, Bucuresti 1991. 3. J. ROTMAN, Advanced modern algebra, Prentice Hall, NJ 2002. 4. G. CALUGAREANU, P. HAMBURG: Exercises in basic ring theory, Kluwer, Dordrecht 1998. 5. S. CRIVEI: Basic Abstract Algebra, Casa Cartii de Stiinta, Cluj-Napoca 2002. 6. A. MARCUS : Algebra [http://math.ubbcluj.ro/~marcus] 7. J. SZENDREI: Algebra es szamelmelet, Tankonyvkiado, Budapest 1993. 8. M. BALINT, G. CZEDLI, A. SZENDREI: Absztrakt algebrai feladatok, Tankonyvkiado, Budapest1988. 9. G. SCHEJA, U. STORCH: Lehrbuch der Algebra 1,2, B.G. Teubner, Stuttgart 1994 10. M. ARTIN: Algebra, Birkhauser, Basel 1998. 11. I. PURDEA, C. PELEA, Probleme de algebra, EFES Cluj-Napoca 2005.
 Assessment Homeworks (20%). Exam (80%).
 Links: Syllabus for all subjects Romanian version for this subject Rtf format for this subject