#### A.Y. 2020 / 2021

Professor
Period
First semester
Credits
6
Duration/Length
48
Type of Learning Activity
Programme specific subjects
Study Path
[PDS0-2018 - Ord. 2018] common
Syllabus
Teaching language

Italian/English

Learning objectives

Learn the explained material

Prerequisites

Having taken the courses of the Laurea triennale in Matematica: Algebra
1,2 or equivalent

Contents

Rings and modules
Fundamental theorem of finite abelian groups
PIDs, Euclidean domains, UFD, artinian and noetherian rings
Primary decomposition of ideals (modules)
Modules over a PID
Dedekind domains

Teaching format

Lessons

Extended Programme

This is a draft of the content of the course:
Basic notions. Direct sum of modules, product of modules, homomor-
phisms, free modules. Abelian groups, fundamental theorem of finite abelian
groups. The Jordan-Holder theorem.
Rings. Principal ideals rings, euclidean rings, unique factorization domains,
artinian and noetherian rings. Primary decomposition of ideals (modules) over
a noetherian ring.
Modules over PID. Generalization of the fundamental theorem of finite
abelian groups. Canonical forms of matrices.
Dedekind domains. Characterization of Dedekind domains. Unique fac-
torization of ideals. Fractional ideals, groups associated to the ideals of a
Dedekind domain.

End-of-course test

Oral examination, evaluated (if positive) from 18/30 (poor) to 30/30 cum
laude (very good).

Other information

Further information on the website:
http://www.dmi.units.it/~logar/didattica/IstAlgSup/

Any changes to the methods described here, which become necessary toensure the application of the safety protocols related to the COVID19emergency, will be communicated on the websites of the Department ofMathematics and Geoscience - DMG and of the Study Program inMathematics.

Texts/Books

Partial bibliography
Atyah, Mac Donald, Introduction to commutative algebra
Birkhoff, Mac Lane, Algebra