Università Roma Tre

1. Modules

Modules and submodules. Operations between submodules. Annihilator. Homomorphisms and quotient modules. Generators and bases. Free modules. Rank invariance.
Direct sum and direct product. Tensor product of
modules. Universal property. Tensor product of algebras. Exactness of
tensor product. Flat modules. Extension and restriction of scalars. The Theorem of Caylay-Hamilton. Nakayama's Lemma.

2. Ideals
Operations between ideals. Homomorphisms of rings and quotient rings. Prime and primary ideals. Zorn's lemma. Maximal and minimal ideals. Jacobson's radical
and Nilradical. Radical ideals. Reduced rings. The Chinese Remainder Theorem. First
Avoidance Theorem. Fractional ideals of domains. Invertible ideals.

3. Rings and modules of fractions
Multiplicative parts. Saturated multiplicative parts. Rings and modules of fractions.
Extension and contraction of ideals. Prime and primary ideals in rings of fractions. Local rings. Local properties. The ring of formal series on a field.


4. Integral dependence
Integral dependence and integral closure. Stability and transitivity properties
of full dependence. Lying over, Inc and Going up. Krull size of the
integral closure. Notes on the noetherian nature of integral closure.
Valutation rings and their characterizations. Discrete valuation rings. Krull theorem on the integral closure. Dedekind Rings

5. Noetherian and Artinian rings and modules
Chain conditions and equivalent properties. Noetherian rings and modules.
Modules and algebras on Noetherian rings. The Hilbert Basis Theorem. Cohen's Theorem. Primary decomposition of ideals. Uniqueness theorems. Associated primes
and zerodivisors. Artinian rings and modules. Artionian Ring characterization theorem. The Principal Ideal Theorem.
6. Completion of rings.

M. F. Atiyah, I. G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley, 1969.

R. Gilmer, Multiplicative Ideal Theory, Dekker, New York, 1972