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
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
Oral examination, evaluated (if positive) from 18/30 (poor) to 30/30 cum
laude (very good).
Further information on the website:
Atyah, Mac Donald, Introduction to commutative algebra
Birkhoff, Mac Lane, Algebra
Rotman, Advanced Modern Algebra