Degree in MATHEMATICSItalian/English
Learn the explained material
Having taken the courses of the Laurea triennale in Matematica: Algebra
1,2 or equivalent
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
Lessons
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.
Oral examination, evaluated (if positive) from 18/30 (poor) to 30/30 cum
laude (very good).
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.
Partial bibliography
Atyah, Mac Donald, Introduction to commutative algebra
Birkhoff, Mac Lane, Algebra
Rotman, Advanced Modern Algebra
