Syllabus

Date Notes Topics
Sep 5 1.1 Refresher on modules
Sep 10 1.2–1.4 Noetherian and artinian modules and rings
Sep 12 1.4, 2.1 Jordan-Hölder Theorem, Affine varieties, vanishing ideals
Sep 17 2.1–2.3 Hilbert Basis Theorem, radical ideals
Sep 19 2.4 Integral extensions
Sep 24 2.5 Noether Normalization Lemma
Sep 26 2.6 Hilbert's Nullstellensatz
Oct 1 Localization
Oct 3 Localization
Oct 10 Tensor product of modules over a commutative ring
Oct 15 Tensor product of modules over a commutative ring
Oct 17 General tensor products
Oct 22–26 Reading week
Oct 29 Semisimple modules, examples, equivalent characterizations of semisimple modules, submodules and quotients of semisimple modules
Oct 31 Schur's Lemma, Jacobson Density Theorem
Nov 5 Definition of semisimple rings, decomposition of a semisimple ring into its basic building blocks
Nov 7 Midterm test
Nov 12 Decomposition of a semisimple ring into its basic building blocks, properties of simple rings
Nov 14 Characterization of simple rings, Artin-Wedderburn Theorem
Nov 19 Short exact sequences, split short exact sequences
Nov 21 Left exactness of Hom, right exactness of the tensor product
Nov 26 Projective modules
Nov 28 Injective modules
Dec 3 Flat modules
Dec 5 Review