Algebraic Geometry: Localization and Prime Ideals

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Affine Varieties and Hypersurfaces

Explores affine varieties, hypersurfaces, dimension in algebraic geometry, minimal prime ideals, and local properties of plane curves.

Primary Decomposition in Rings

Explores primary decomposition in rings, focusing on primary ideals and their properties.

Local Rings and Residues

Covers the proof of theorem 4.2 on multiplicities and the special structure of local rings at a simple point of a plane.

Affine Algebraic Varieties: Zariski Topology

Explores affine algebraic varieties, emphasizing the Zariski Topology and regular functions.

Prime ideals and local rings

Covers the concept of prime ideals and local rings, emphasizing their importance.

Group Algebra: Maschke's Theorem

Explores Wedderburn's theorem, group algebras, and Maschke's theorem in the context of finite dimensional simple algebras and their endomorphisms.

Irreducible Algebraic Sets

Explores irreducible algebraic sets and their unique decomposition into components, with a focus on prime ideals and plane subsets classification.

Mathematics: Sets and Functions

Introduces sets, functions, Cartesian products, and compositions, discussing images, preimages, and function properties.

Galois Theory: Dedekind Rings

Explores Galois theory with a focus on Dedekind rings and their unique factorization of fractional ideals.

Primary Decomposition: Understanding Schemes

Explores primary decomposition and schemes in algebraic geometry, emphasizing the importance of working over non-algebraically closed fields and the concept of fibers of morphisms.