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 of Ideals Prime avoidance, associated primes, and their properties 7 Primary Decomposition Primary decomposition and uniqueness for modules over Noetherian rings 8 Integral Extensions The Cayley-Hamilton theorem, integral elements, and integral extensions 9 Integral Extensions and Ideals Lying over and going up theorems for prime ideals in integral extensions 10 Nullstellensatz Proofs of Jacobson rings and Hilbert’s Nullstellensatz 11 Blowup Algebras Rees algebra and associated graded ring from an ideal 12 Flatness Definition of flat modules, characterization via Tor, and basic properties 13 Flatness and Localization A local criterion for flatness via checking at the maximal ideal 14 Completion Completion of rings and modules defined by a filtration 15 Properties of Completion Compatibility of completion with exact sequences, Artin-Rees lemma 16 Dimension Krull dimension, defined by prime chains, and its basic properties 17 System of Parameters The relationship between the system of parameters of a local ring and dimension 18 Fractional Ideals Fractional ideals, invertible modules, and the Picard group 19 Regular Local Rings Characterization of regular systems of parameters and regular local rings 20 Divisors Cartier divisors and class groups in Dedekind domains 21 Noether Normalization Noether normalization theorem and applications for finitely generated algebras 22 Differentials The algebraic definition of the Kähler differential module and its universal property