Università Roma Tre

Rings and ideals, maximal ideals and prime ideals, nilradical andJacobson radical, the 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. Primary decomposition. Integral dependence and valuation. Chain conditions. Noetherian rings, Hilbert's Basis Theorem, Nullstellensatz. Discrete valuation rings and Dedekind domains. Hints of dimension theory.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra. Addison-Wesley, 1996.
M. Reid, Undergraduate Algebraic Geometry, Cambridge University Press, 1988.
D. Eisenbud, Commutative Algebra with a view toward Algebraic Geometry, Springer-Verlag, 1995.
A. Gathmann, Commutative Algebra, Lecture notes.