Review of rings, ideals, modules, submodules, tensor product of modules, Noetherian rings.
Operations on modules, exact sequences, Adjointness of Hom and Tensor.
Localization and Spec of a ring.
Associated primes and primary decomposition.
Artinian rings, Krull dimension and Hilbert’s Nullstellensatz.
Integral extensions, Going up, Noether’s Normalization, Going down.
Discrete valuation rings and Dedekind domains.
Graded rings and graded modules, Artin-Rees Lemma, Krull’s Intersection Theorem.