Summaries

Lecture 6 (12/10) - Graded version of Nakayama's lemma. Noetherian graded rings. Proof that the associated primes of a graded ideal are also graded. Associated graded rings and modules. The Artin-Rees lemma and Krull's Intersection theorem.  (Sections 3.1 and 3.3)

Lecture 5 (05/10) - Primary ideals. Primary decompositions of ideals in Noetherian rings and the first and second uniqueness theorem for irredundant primary decompositions. Introducing graded rings, graded modules and graded submodules. (Sections 2.4 and 2.5. Start of Section 3.1)

Lecture 4 (28/09) - Minimal primes. Associated primes. The set of zero divisors as the union of the associated primes of the module. Prime filtrations of finitely generated modules over Noetherian rings. (Sections 2.2 and 2.3)

Lecture 3 (21/09) - The radical of an ideal as the intersection of the primes that contain the ideal. Length. Artinian rings and modules. (Sections 1.8 and 2.1)

Lecture 2 (14/09) - Prime avoidance. Localization. Characterizing the ideals and the prime ideals in the localized ring. Noetherian rings and modules. (Sections 1.4, 1.5 and 1.7)

Lecture 1 (07/09) - Introducing the basic concepts of commutative rings, modules and ideals. Prime and maximal ideals. Local rings and Nakayama's lemma. (Sections 1.1, 1.2 and 1.3)