First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCL1001443, A.A. 2018/19

Information concerning the students who enrolled in A.Y. 2018/19

Information on the course unit
Degree course Second cycle degree in
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2018/19
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Degree course track GENERALE [010PD]
Number of ECTS credits allocated 6.0
Type of assessment Mark
Course unit English denomination RINGS AND MODULES
Website of the academic structure
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English
Single Course unit The Course unit can be attended under the option Single Course unit attendance
Optional Course unit The Course unit can be chosen as Optional Course unit

Teacher in charge SILVANA BAZZONI MAT/02

Course unit code Course unit name Teacher in charge Degree course code

ECTS: details
Type Scientific-Disciplinary 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 of
Individual study
Practice 2.0 16 34.0 No turn
Lecture 4.0 32 68.0 No turn

Start of activities 25/02/2019
End of activities 14/06/2019

Examination board
Board From To Members of the board
7 Anelli e Moduli - a.a. 2018/2019 01/10/2018 30/09/2019 BAZZONI SILVANA (Presidente)
COLPI RICCARDO (Membro Effettivo)

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. Freyd-Mitchell embedding theorem. Pull-back and push-out. 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, Springer-Verlag, 1975. Cerca nel catalogo
  • C.A. Weibel, An Introduction to Homological Algebra. --: Cambridge studies in Ad. Math., 38, 1994. Cerca nel catalogo
  • J. Rotman, An introduction to Homological Algebra. New York: Universitext Springer, 2009. Cerca nel catalogo

Innovative teaching methods: Teaching and learning strategies
  • Questioning
  • Loading of files and pages (web pages, Moodle, ...)

Innovative teaching methods: Software or applications used
  • Moodle (files, quizzes, workshops, ...)
  • One Note (digital ink)

Sustainable Development Goals (SDGs)
Quality Education Gender Equality Reduced Inequalities