Università Roma Tre

Rings and ideals, maximal ideals and prime ideals, nilradical and Jacobson radical, spectrum of a ring.
Modules, finitely generated modules and Nakayama's Lemma, exact sequences, tensor product, restriction and extension of scalars.
Rings and modules of fractions, localization. Composition series and length of a module. Chain conditions. Noetherian rings, Hilbert's Basis Theorem. Integral extensions, Lying Over and Going-up theorems. Noether normalization theorem and Hilbert's Nullstellensatz.
Krull dimension and Krull's principal ideal theorem. Transcendence degree. Dimension of local Noetherian rings. Regular rings.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.
A. Gathmann, Commutative Algebra, Lecture notes.
A. Chambert-Loir, (Mostly) Commutative Algebra, Springer Cham, 2021