First cycle
degree courses
Second cycle
degree courses
Single cycle
degree courses
School of Science
Course unit
SCP3050935, A.A. 2017/18

Information concerning the students who enrolled in A.Y. 2017/18

Information on the course unit
Degree course Second cycle degree in
SC1172, Degree course structure A.Y. 2011/12, A.Y. 2017/18
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Degree course track GENERALE [010PD]
Number of ECTS credits allocated 8.0
Type of assessment Mark
Course unit English denomination COMMUTATIVE ALGEBRA
Website of the academic structure
Department of reference Department of Mathematics
Mandatory attendance No
Language of instruction English


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

ECTS: details
Type Scientific-Disciplinary Sector Credits allocated
Core courses MAT/03 Geometry 8.0

Mode of delivery (when and how)
Period First semester
Year 1st Year
Teaching method frontal

Organisation of didactics
Type of hours Credits Hours of
Hours of
Individual study
Practice 4.0 32 68.0 No turn
Lecture 4.0 32 68.0 No turn

Start of activities 02/10/2017
End of activities 19/01/2018

Prerequisites: Basic notions of algebra (groups, rings, ideals, fields, quotients, etc.), as acquired in the "Algebra 1" course.
Target skills and knowledge: A good knowledge of the algeraic objects used in Algebraic Geometry and Number Theory:
- Modules;
- Tensor products;
- Prime spectrum;
- Localization;
- Integral extensions;
- Noetherian rings;
- Dedekind domains and discrete valuation rings;
- Basics on dimension theory.
Examination methods: Written exam
Assessment criteria: The student will be evaluated on his/her understanding of the topics, on the acquisition of concepts and methodologies proposed and on the ability to apply them in full independence and awareness.
Course unit contents: Commutative rings with unit, ideals, homomorphisms, quotient rings. Fields, integral domains, zero divisors, nilpotent elements. Prime ideals and maximal ideals. Local rings and their characterization. Operations on ideals (sum, intersection, product). Extension and contraction of ideals w.r.t. homomorphisms. Annihilator, radical ideal, nilradical and Jacobson radical of a ring. The Zariski topology on the prime spectrum Spec(R). Spec(R/I) as closed subset of Spec(A). Direct product of rings.

Modules, submodules and their operations (sums, intersection). Annihilator of a module. Faithful modules. Direct sums and direct products of modules. Exact sequences of modules, snake lemma. Projective and injective modules. Finitely generated and finitely presented modules, free modules. Cayley-Hamilton theorem and Nakayama's lemma.

Tensor product and its properties. Extension of scalars for modules. Algebras over a ring and their tensor product. Adjunction and exactness of the Hom and tensor product functors. Flat modules. Kahler differentials

Rings of fractions and localisation. Exactness of localisation. of rings and modules. Localisation and open subsets of Spec(R). Local properties. faithfully flat modules and descent theory. Projective and locally free modules.

Integral elements, integral extension of rings and integral closure. Going Up, Going Down and geometric translation. Norm, trace, discriminant. Valuation rings. Overview of completions.

Chain conditions, Artinian and Noetherian rings and modules. Hilbert's basis theorem. Normalization Lemma and Nullstellensatz.

Discrete valuation rings. Fractional ideals and invertible modules. Cartier and Weil divisors, Picard group, cycle map. Dedekind domains and their extensions. Decomposition of ideals, inertia, ramification.

Krull dimension, height of a prime ideal. Principal ideal theorem. Characterisation of factorial domains. Regular local rings. Finiteness of dimension for local noetherian rings.
Planned learning activities and teaching methods: Lectures. Recommended exercises.
Additional notes about suggested reading: Lecture notes available at
Textbooks (and optional supplementary readings)
  • Garuti, M.A., Commutative Algebra Lecture notes. Padova: --, 2015. Disponibile gratuitamente alla pagina web dell'autore.
  • Atiyah, Michael Francis; Mac_Donald, Ian Grant, Introduction to commutative algebraM. F. Atiyah, I. G. Macdonald. Reading [etc.]: Addison-Wesley, --.
  • Atiyah, Michael Francis; Mac_Donald, Ian Grant; Maroscia, Paolo, Introduzione all'algebra commutativaM. F. Atiyah e I. G. Macdonaldappendice all'edizione italiana di Paolo Maroscia. Milano: Feltrinelli, 1981.
  • Eisenbud, David, Commutative algebra with a view toward algebraic geometry. New York [etc.]: Springer, --. Cerca nel catalogo
  • Gathmann, A., Commutative Algebra. Kaiserslautern: --, 2013. Disponibile gratuitamente alla pagina web dell'autore.