Unit COMMUTATIVE AND COMPUTATIONAL ALGEBRA
- Course
- Mathematics
- Study-unit Code
- 55A00034
- Curriculum
- Matematica per l'economia e la finanza
- Teacher
- Giuliana Fatabbi
- Teachers
-
- Giuliana Fatabbi
- Hours
- 63 ore - Giuliana Fatabbi
- CFU
- 6
- Course Regulation
- Coorte 2024
- Offered
- 2025/26
- Learning activities
- Caratterizzante
- Area
- Formazione teorica avanzata
- Academic discipline
- MAT/02
- Type of study-unit
- Opzionale (Optional)
- Type of learning activities
- Attività formativa monodisciplinare
- Language of instruction
- Italian
- Contents
- An in-depth study of commutative rings with unity, modules over commutative rings, and an introduction to Gröbner bases theory. Special emphasis will be placed on computational methods and on applications of commutative algebra to algebraic geometry.
- Reference texts
- Atiyah-Macdonald, Intoduction to Commutative algebra, Addison-Wesley, 1969
Cox-Little-O'Shea, Ideals, Varieties, and Algorithms, Springer , 1997
Any supplementary material available in Unistudium - Educational objectives
- The course aims to: provide an in-depth study of commutative rings with unity, focusing particularly on polynomial rings and their quotients, with an eye toward applications in algebraic geometry; introduce the theory of Gröbner bases as a gateway to computational algebra and its main applications. The course also aims to enhance students' abstract reasoning skills while illustrating how strong theoretical foundations can lead to the development of meaningful computational tools. Expected learning outcomes Knowledge and understanding: Knowledge of fundamental results and methods in the theory of algebraic structures and their applications. Ability to read, comprehend, and deepen understanding of topics from mathematical literature and present them clearly and rigorously. Ability to analyze mathematical problems by identifying their essential components. Applying knowledge and understanding: Ability to construct and solve examples and exercises, tackle new theoretical problems, and apply suitable techniques effectively. Development of computational skills, including proficiency with software tools for computational algebra. Making judgements: Ability to construct rigorous logical arguments, clearly distinguishing between assumptions and conclusions. Ability to recognize valid proofs and identify flawed reasoning. Capability to independently assess the applicability of algebraic models to both theoretical and practical problems. Communication skills: Ability to clearly and accurately present mathematical concepts, problems, and solutions—both one’s own and others’—in oral and written form, adapting language to the audience. Ability to clearly justify methodological and computational choices. Learning skills: Ability to study and explore advanced topics in algebraic literature independently. Readiness to systematically learn new content in commutative and computational algebra.
- Prerequisites
- Fundamental concepts of rings and ideals, with particular focus on polynomial rings over a field.
- Teaching methods
- Face to face lessons
- Other information
- According with the students attending the course, the course can be partially or entirely given in English. The exam can be in English, upon request of the student.
Usage of platform "uni-studium" - Learning verification modality
- Examination method The final examination consists of an oral test lasting approximately 45–60 minutes. During the exam, students will be asked to: present the solution to 2 or 3 exercises assigned at the end of the course; explain and discuss selected topics covered during the lectures. The oral exam is intended to assess the student's level of understanding, critical thinking, and personal engagement with the course content. Upon request, the exam may be conducted in English. Accessibility and support For information regarding support services for students with disabilities and/or specific learning disorders (SLD), please visit: www.unipg.it/disabilita-e-dsa. The instructor is available to assess on a case-by-case basis the need for compensatory measures or personalized learning paths for students with disabilities, SLD, working students, or non-attending students.
- Extended program
- I. Rings and Ideals Basic properties of commutative rings with unity Prime and maximal ideals Local rings Nilradical and Jacobson radical Operations on ideals; radical of an ideal Ring homomorphisms Extension and contraction of ideals II. Modules Definition and fundamental properties Direct sums, direct products; free modules Finitely generated modules and Nakayama’s Lemma Module homomorphisms Algebras over a ring III. Rings and Modules of Fractions Definition and general properties Localization and local properties Ideals in rings of fractions IV. Noetherian Rings and Algebraic Geometry Definition and properties of Noetherian rings Affine varieties and affine ¿ K-algebras The basic dictionary between algebra and geometry Krull dimension Noetherian modules: definitions and key properties V. Artinian Rings Definitions and properties Relationship between Artinian and Noetherian rings Characterization: a ring is Artinian if and only if it is Noetherian and of dimension zero VI. Primary Decomposition Primary ideals and their properties Primary decomposition Associated primes, zero divisors Uniqueness of isolated components The Noetherian case VII. Hilbert's Nullstellensatz Weak and strong forms VIII. Integral Dependence and Normal Domains Definitions and fundamental properties The Going Up Theorem Normal integral domains and the Going Down Theorem IX. Elements of Dimension Theory Chains of prime ideals, height, and dimension Krull’s Principal Ideal Theorem and Height Theorem Dimension of polynomial rings over a field Local rings: systems of parameters, embedding dimension Local regular rings (definition and geometric significance only) X. Gröbner Bases: Basic Theory Gröbner bases in the linear and univariate case Monomial orderings Division algorithm Definition of Gröbner bases S-polynomials and Buchberger’s Algorithm Reduced Gröbner bases XI. Applications of Gröbner Bases Elementary applications of Gröbner bases Elimination theory Polynomial maps Selected applications to algebraic geometry