INSTITUTIONS OF HIGHER ALGEBRAModule MODULO I
Academic Year 2025/2026 - Teacher: Marco D'ANNAExpected Learning Outcomes
The aim of this course is to deepen the study of coomutative ring theory, taking particular attention to polynomial rings and their quotients, with a view towards applications to allgebraic geometry and number theory.
The student will also improve his skill of making abrstact argomentations and will learn that a deep theoretical knowledge allows to develop signifinant applicative tools.
Course Structure
The course consists of lectures, frontal exercises (on the blackboard) and class exercises.
Normally the exercises done by the teacher alternate with the theoretical part, even
on the same day. For the class exercises, the teacher proposes some exercises to the
students who are invited to solve them working in small groups; the teacher goes between
the desks helping and suggesting the way to tackle the exercises. These exercises are
essential to acquire the ability to work independently and in a group.
Information for students with disabilities and/or DSA. To ensure equal opportunities
and in compliance with current laws, interested students may request a personal
interview in order to plan any compensatory and/or dispensatory measures, based on
their educational objectives and specific needs. Students with disabilities and/or
DSA must contact the teacher, the CInAP contact person of the DMI (Prof. Daniele)
and CInAP well in advance of the exam date to communicate that they intend to take
the exam using the appropriate compensatory measures (which will be indicated by CInAP).
Required Prerequisites
Knowledge of basic algebra (sets, numbers, polynomials, rings, groups), as required
in the requirements of the Master's Degree in Mathematics.Attendance of Lessons
Highly recommended. Attending lessons and exercises allows the student to integrate
the theory presented in the reference texts and to learn how to correctly set up
the exercises independently.Detailed Course Content
I. Rings and ideals. Definitions and first properites. Prime and maximal ideals. Local rings. Nilradical and Jacobson radical. Ideals operations.Homomorphisms.
II. Modules. Definitions and first properties. DIrect product and direct sum; free modules. Finite modules and Nakayama's lemma. Module homomorphisms. Algebras.
III. Frctions rings and modules. Definition and properties. Localization and local properties. Ideals in fraction rings.
IV. Noetherian rings. Affine varieties, affine K-algebras, correspondance betwwen algebra and algebraic-geometry concepts. Krull dimension. Noetherian rings and modules: definitions and first properties. Hilbert's basis theorem. Conditions for a sub-algebra to be finitely generated.
V. Artinian rings. Artinian rings and modules. Composition series. Length. A ring is artinian if and only if it is noetherian and zero-dimensional.
VI. Primary decomposition. Primary ideals and primary decomposition. Associated primes and their characterization. Zero-divisors. Unicity of isolated components. The noetherian case.
VII. Hilbert Nullstellensatz: weak and strong formulations.
VIII. Integral dependance. Definitions and first properties. Going Up theorem. Normal domains and Going Down theorem. Noether's normalization lemma.
IX. First steps in dimension theory. Chain of primes, height, dimension. Krull's principal ideal theorem.Krull's height theorem.Dimension for polynomial rings with coefficient in a field. Local rings. System of parameters. Embedding dimension. Regular local rings (only definition and geometric relevance).
Textbook Information
1. M.F. Atiyah, I.G. Macdonald , Introduzione all' algebra commutativa, Feltrinelli 1981
2. E. Kunz, Introduction to commutative algebra and algebraic geometry, Birkauser 1985
Course Planning
| Subjects | Text References | |
|---|---|---|
| 1 | Rings and ideals: First properties of commutative rings with units. Prime ideals and maximal ideals. Local rings. Nilradical and Jacobson radical. | 1 |
| 2 | Operations with ideals; radical of an ideal. Homomorphisms. Extended ideals and contracted ideals. | 1 |
| 3 | Modules: definition and first properties. Direct product and direct sum: free modules. Finitely generated modules and Nakayama's lemma. Homomorphisms between modules. Algebras. | 1 |
| 4 | Fraction rings and moduli: definition and properties. Localization and local properties. Ideals in fraction rings. | 1 |
| 5 | Affine varieties, affine K-algebras and basic algebra-algebraic geometry dictionary. Krull dimension. | 2 |
| 6 | Noetherian rings and modules: definitions and first properties. Hilbert's basis theorem. Conditions for a subalgebra to be finitely generated. | 1 |
| 7 | Artinian rings and modules. Composition series. Length. A ring is Artinian if and only if it is Noetherian and has zero dimension. | 1 |
| 8 | Primary ideals; primary decomposition. Associated primes and their characterization. Divisors of zero. Uniqueness of isolated components. The Noetherian case. | 1 |
| 9 | Hilbert's zero theorem: weak form and strong form. | 1 or 2 |
| 10 | Integral dependence: definitions and first properties. Going Up Theorem. Normal domains and Going Down Theorem. | 1 |
| 11 | Noether's normalization lemma. | 1 or 2 |
| 12 | Prime chains, height, dimension. Krull's principal ideal theorem. Krull's height theorem. | 1 |
| 13 | Dimension of rings of polynomials with coefficients in a field. Local rings. Parameter system. Embedding dimension. Regular local rings(definition and geometric importance only). | 1 |
Learning Assessment
Learning Assessment Procedures
During the semester, exercises will be assigned and in-class exercises will be carried out,
during which students will be invited to try, alone or in collaboration, problems proposed by the
teacher. Correcting the exercises assigned for homework and
the in-class exercises will allow the teacher to verify the level of understanding of the subject.
There will be two written tests in progress for the Commutative Algebra module.
At the end of the course (or module) there will be an oral test that will take into account
the exercises completed during the year and the in-class tests. During this test, the student
will be asked, with regard to the Commutative Algebra module, to solve exercises related to
the program covered and to illustrate some theoretical topics.
The learning assessment may also be carried out electronically, if conditions require it.Examples of frequently asked questions and / or exercises
Algebra exercises are not standard and therefore it is not possible to describe
typologies of problems. Exercises related to the various topics of the course will
be made available on Studium. The structure of a theoretical question is the following:
illustrate a topic, state the definitions and the main results inherent to that topic,
demonstrate one of them and bring examples and counterexamples.