
Course unit
RINGS AND MODULES
SCL1001443, A.A. 2019/20
Information concerning the students who enrolled in A.Y. 2019/20
Mutuating
Course unit code 
Course unit name 
Teacher in charge 
Degree course code 
SCL1001443 
RINGS AND MODULES 
SILVANA BAZZONI 
SC1172 
ECTS: details
Type 
ScientificDisciplinary Sector 
Credits allocated 
Core courses 
MAT/02 
Algebra 
6.0 
Course unit organization
Period 
Second semester 
Year 
1st Year 
Teaching method 
frontal 
Type of hours 
Credits 
Teaching hours 
Hours of Individual study 
Shifts 
Practice 
2.0 
16 
34.0 
No turn 
Lecture 
4.0 
32 
68.0 
No turn 
Prerequisites:

Notions from the Algebra courses of the first two years of the degree in Mathematics and basic notions on module theory over arbitrary rings. 
Target skills and knowledge:

The aim of the course is to learn the basic notions in catgeory theory and the related main constructions. To introduce the techniques and the tools of homological algebra and their applications to dimension theory. 
Examination methods:

Written exam consisting in answering to questions from the theory and in solving exercises.
Discussion of the composition and possible oral exam. 
Assessment criteria:

Check of the learning of the taught notions and on the ability of their application. 
Course unit contents:

Additive and Abelian categories. Functor categories. FreydMitchell embedding theorem. Pullback and pushout. Limits and colimits. Adjoint functors. Categories of chain complexes and the homotopy category. Foundamental Theorem in homology. Left and right derived functors. The functors Tor, flatness and purity. The funtors Ext and Yoneda extensions. Flat, projective and injective dimensions of modules and their characterization in terms of derived functors. Applications to the global dimension of rings and Hilbert's syzygies Theorem. 
Planned learning activities and teaching methods:

Lists of exercises to solve will be distributed to check and to deepen the learning of the taught notions.
The notes of the lectures will be distributed daily. 
Additional notes about suggested reading:

Notes of the lectures, solving of the proposed exercises. Reading of the books in the bibliography. 
Textbooks (and optional supplementary readings) 

B.B Stentrom, Rings of quotients. : Grundleheren der Math., 217, SpringerVerlag, 1975.

C.A. Weibel, An Introduction to Homological Algebra. : Cambridge studies in Ad. Math., 38, 1994.

J. Rotman, An introduction to Homological Algebra. New York: Universitext Springer, 2009.


