Babes-Bolyai University of Cluj-Napoca
Faculty of Mathematics and Computer Science
Study Cycle: Master
SUBJECT
| MML1002 | Module Theory |
| 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 |
| References |
|
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. |
| Links: | Syllabus for all subjects Romanian version for this subject Rtf format for this subject |