Lecture
Regularity and Integrality
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Regular Implicit Functions
Discusses regular implicit functions, rings, ideals, sheaves, and ring operations.
Simple Modules: Schur's Lemma
Covers simple modules, endomorphisms, and Schur's lemma in module theory.
Principal Ideal Domains: Structure and Homomorphisms
Covers the concepts of ideals, principal ideal domains, and ring homomorphisms.
Primary Decomposition in Rings
Explores primary decomposition in rings, focusing on primary ideals and their properties.
Rings and Modules
Covers rings, modules, fields, minimal ideals, and the Nullstellensatz theorem.
Affine Varieties and Hypersurfaces
Explores affine varieties, hypersurfaces, dimension in algebraic geometry, minimal prime ideals, and local properties of plane curves.
Congruence Relations in Rings
Explores congruence relations in rings, principal ideals, ring homomorphisms, and the characteristic of rings.
Idempotent Elements and Central Orthogonal
Explores idempotent elements, central orthogonal elements, commutative rings, and prime ideals in non-central rings.
Irreducible Factors and Noetherian Rings
Explores irreducible factors, Noetherian rings, ideal stability, and unique factorization in rings.
Additivity of Dimension & Height
Explores the additivity of dimension and height in a finitely generated k-algebra domain, showcasing its applications and implications.