Lectures related to The Legacy of Rome: Honor and Freedom

Ring Operations: Ideals and Classes

Covers the operations in rings, ideals, classes, and quotient rings.

Ideals and PPCM

Covers the concept of ideals in polynomial rings and their properties.

Algebraic Subsets of A^1

Covers algebraic subsets of A^1 and ideals with a finite set of points.

Fractions of Rings

Explores the concept of fractions of rings and their uniqueness in ideals and quotients.

Hilberts Nullstellensatz and Ideals

Explores ideals with finite sets of points and Hilberts Nullstellensatz in algebraic fields.

Regular Implicit Functions

Discusses regular implicit functions, rings, ideals, sheaves, and ring operations.

Primary Decomposition: Noetherian Ideal

Covers primary decomposition in Noetherian ideals and unique primary ideals.

Chinese Idempotent Constructions and Properties

Explores Chinese idempotent constructions and their properties, including ideals and product characterization.

Cyclotomic Extensions: Norms, Ideals, and Primes

Explores cyclotomic extensions, prime numbers, and ideal norms in number theory.

Ideals and Representations

Covers ideals, representations, modules, and maximal ideals in associative algebras.

Generators of Ideals

Explores the calculation of generators of ideals and their significance in mathematical operations.

Dedekind Rings and Fractional Ideals

Explores Dedekind rings, fractional ideals, integrally closed properties, prime ideal factorization, and the structure of fractional ideals as a commutative group.

Principal Ideal Domains: Structure and Homomorphisms

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

Ring Constructions: Structure Theorems

Explores operations on ideals and structure theorems in commutative rings.

Dimension Theory of Rings

Explores the dimension theory of rings, focusing on chains of ideals and prime ideals.

Prime ideals and local rings

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

Polynomials on a Field: Properties and Applications

Explores the properties and applications of polynomials on a field, including formal derivation and uniqueness.

Localization and Ideals

Covers the concept of localization and ideals in rings, focusing on extended and contracted ideals, integral extensions, and monic equations.

Primary and Secondary Ideals

Covers the concept of primary and secondary ideals, focusing on uniqueness and invariance properties.

Primary Decomposition in Rings

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