COMMUTATIVE ALGEBRA

The aim of this course is to deepen the study of commutative ring theory and to provide a suitable background toward advanced studies in Commutative Algebra and Geometry. 

The student will also improve his skill of making abrstact argomentations and will learn that a deep theoretical knowledge allows to develop signifinant applicative tools.

I. Rings and ideals. Prime and maximal ideals. Local rings. Nilradical and Jacobson radical of a ring. Ideal operations. Radical of an ideal. Extension and contraction of an ideal. Prime spectrum of a ring. 

II. Modules. First properties of modules. Direct product and direct sum. Finitely generated modules and Nakayama's Lemma. Free modules. Algebras.

III. Rings and modules of fractions. First properties. Localizations and local properties. Ideals in rings of fractions. 

IV. Integral dependence. First properties of integral dependence. Going up and Lying over theorems. Normal domains and the Going Down theorem. 

V. Noetherian rings. Noetherian rings and Noetherian modules: first properties. The Hilbert basis Theorem and its consequences. 

VI. Artinian rings. Anelli e moduli artiniani. Serie di composizione. Lunghezza. Un anello è artiniano se e soltanto se è noetheriano e ha dimensione zero. Artinian rings and modules. Composition series. Length. A ring is Artinian if and only if it is Noetherian and zero-dimensional. 

VII. Primary decomposition. Primary ideals. Primary decomposition. Associated primes and their characterization. Zero divisors. Unicity of isolated components. 

VIII. The Hilbert's Nullstellensatz: proof given by Goldman and Krull. 

IX. First steps in dimension theory. Chains of prime ideals. Height. dimension. Hilbert's Hauptidealsatz. Dimension of polynomial rings over a field. 

X. Introduction to valuation theory. Valutazioni su un campo e loro anelli di valutazione. Teoremi di dominanza. Globalizzazione: domini di Dedekind e di Prufer. Valuations over a field and their valuation rings. Domination theorems. Globalization: Dedekind domains and Krull domains. 

XI. Introduction to topological rings. Topological rings and their properties. Completions. 

2. R. Gilmer, Multiplicative ideal theory. Pure and Applied Mathematics, No. 12. Marcel Dekker, Inc., New York, 1972.

3. I. Kaplansky, Commutative rings, University of Chicago Press, Chicago, Ill.-London, 1974. 

4. O. Zariski, P. Samuel, Commutative algebra. Vol. 1. Graduate Texts in Mathematics, No. 28. Springer-Verlag, New York-Heidelberg-Berlin, 1975.

5. O. Zariski, P. Samuel, Commutative algebra. Vol. II.  Graduate Texts in Mathematics, Vol. 29. Springer-Verlag, New York-Heidelberg, 1975.