Algebra Commutativa 2021/2022
- Dipartimento: Scienze Matematiche, Fisiche E Naturali
- Tipologia: Corso Di Laurea Magistrale Dm.270/04
- Corso di Laurea: Matematica Pura Ed Applicata
- Settore Ministeriale: MAT/02
- Codice di verbalizzazione: 8067262
- Metodi di insegnamento: Frontale
- Metodi di valutazione: Orale
- Prerequisiti: The student is required to know - at least to a good extent: possible gaps, if any, can be reasonably amended during the course - the following topics: rings (the basic theory), vector spaces, basics of general topology (including connectedness and compactness).
- Obiettivi: LEARNING GOALS: Achieve a good knowledge of the main structures and results of Commutative Algebra, including the following topics: - (commutative unital) rings, modules over rings, morphisms of rings and of modules; - tensor product among modules; algebras; rings and modules of fractions; - chain conditions for modules and for rings, Noetherian modules and rings, Artinian modules and rings; - the prime spectrum of a ring; - foundations of category theory. KNOWLEDGE AND UNDERSTANDING: The student will have to know the main types of algebraic structures under study, as well as examples and counterexamples which show the various cases and the differences among them; in particular, they will have to understand and apply the key ideas underlying the main notions, and how these apply in different contexts. The theory of categories will be the overall framework within which the student will have to recognize that a unique abstract language grasp a wide variety of specific instances as special cases, that then can be dealt with from a common point of view. APPLYING KNOWLEDGE AND UNDERSTANDING: The student will have to be able to solve problems and exercises concerning the topics dealt with during the course, with professional language and ability to present and defend their arguments with suitable examples. MAKING JUDGEMENTS: The student will have to be able to recognize by themselves when a mathematical problem can be placed in the framework of either one of the theories studied in the course. Moreover, they will have to be able to work starting with the given, fundamental ingredients, but then adding later by themselves (by means of independent research of what is known in literature, or by personally solving specific problems) the necessary tools to cope with the problems under exam. COMMUNICATION SKILLS: The student will have to be able to explain from start to end the arguments we deal with, both in oral form and in written form, and also in mixed mode (i.e., oral form backed by written formulas and/or calculation and/or images); in particular we expect that they might be able to adapt the level of their exposition to different demands - such as detailed or synthetic treatise, reasoned review focusing onto critical points, etc. LEARNING SKILLS: The student will have to develop those learning skills that enable them to carry on studying in a mainly autonomous and self-organised manner, even though with the help of some directions - such as key topics to look for in their search for suitable bibliographical sources.
- Ricevimento: per appuntamento
- A.A.: 2021/2022
- Canale: UNICO
- Crediti: 8
- Obbligo di Frequenza: No