Algebraic Geometry: Localization and Prime Ideals

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Rings and Modules

Covers rings, modules, fields, minimal ideals, and the Nullstellensatz theorem.

Symbolic Powers: Interpretation and Applications

Covers the interpretation and applications of symbolic powers in algebraic structures, focusing on Krull's Hauptideal Satz and Noetherian rings.

Chinese Remainder Theorem: Rings and Fields

Covers the Chinese remainder theorem for commutative rings and integers, polynomial rings, and Euclidean domains.

Module Theory: Definitions and Examples

Introduces the definition and examples of A-modules, including sub-modules and ideals.

Ring Homomorphisms and Ideals

Explores ring homomorphisms, bilateral ideals, ring features, and ideal operations.

Factorisation in Dedekind Ring

Explains the factorisation of ideals in a Dedekind ring using prime ideals and covers ramification index, residual fields, inertia degree, and properties of Dedekind rings.

Principal Ideal Domains: Structure and Homomorphisms

Covers the concepts of ideals, principal ideal domains, and ring homomorphisms.

Ideals in Commutative Rings

Covers the concept of ideals in commutative rings and their role in ring homomorphisms.

Decomposition of Group Algebras

Covers examples of group algebra decomposition and simple infinite-dimensional algebras.

Ring Constructions: Structure Theorems

Explores operations on ideals and structure theorems in commutative rings.