1 Basic Notions Basic conventions and definitions of rings and algebras in commutative algebra 2 Localization Localization of rings and modules, and local ring construction 3 Properties of Localization Compatibility of localization with Hom and tensor, and local properties 4 Localization of Graded Rings Homogeneous localization of graded rings and graded modules 5 The Jordan-Hölder Theorem Uniqueness of composition series and well-definedness of length 6 Associated Primes The prime avoidance lemma and the definition and properties of associated primes 7 Integral Extensions The Cayley-Hamilton theorem, integral elements, and integral extensions 8 Integral Extensions and Ideals The lying over and going up theorems for integral extensions and prime ideals 9 Nullstellensatz Proofs of Jacobson rings and Hilbert’s Nullstellensatz 10 Blowup Algebras Rees algebra and associated graded ring from an ideal 11 Flatness Definition of flat modules, characterization via Tor, and basic properties 12 Flatness and Localization A local criterion for flatness via checking at the maximal ideal 13 Completion Completion of rings and modules defined by a filtration 14 Properties of Completion Compatibility of completion with exact sequences, Artin-Rees lemma 15 Dimension Krull dimension, defined by prime chains, and its basic properties 16 System of Parameters The relationship between the system of parameters of a local ring and dimension 17 Regular Local Rings Characterization of regular systems of parameters and regular local rings 18 Fractional Ideals Fractional ideals, invertible modules, and the Picard group 19 Divisors Cartier divisors and class groups in Dedekind domains 20 Noether Normalization The Noether normalization theorem for finitely generated algebras and its applications