Commutative Algebra.
Commutative Algebra is the "algebra" in Algebraic Geometry: these are two very large research areas that are deeply connected and feed off of each other.
This course will cover the basic fundamentals of Commutative Algebra: chain conditions, Noetherian rings and Hilbert Basis Theorem, primary decomposition,
Noether normalization, Hilbert's Nullstellensatz, Dimension Theory. Regular sequences and Cohen-Macaulay rings, Gorenstein rings, regular rings, complete intersections.
This is most of the first 21 sections of the course textbook, Matsumura's Commutative Ring Theory.
This course is intended as a follow-up to the Homological Algebra course taught this Fall, but this is not a strict prerequisite (please come talk to me in this case).