# Topics - Unital commutative rings, ideals, Hilbert’s basis theorem - Tensor products, flatness, localization - Nakayama’s lemma, integral and finite extensions, going up and going down theorems - Noether’s normalization theorem and Hilbert’s nullstellensatz, primary decomposition - Direct and inverse limits, completions, the Artin-Rees lemma - Dimension theory and Hilbert functions - Discrete valuation rings and Dedekind domains