Lectures related to Ring theory

Fermat's Equation: Integral Solutions and Gaussian Integers

Discusses Fermat's equation and Gaussian integers' properties.

Algebraic Structures of Morphism Spaces

Explores algebraic structures of morphism spaces, including stability properties and injective ring homomorphisms.

Ring Theory: Sub-rings and Morphisms

Introduces sub-rings and morphisms in ring theory, emphasizing stability by multiplication.

Class Number Formula

Explores the Class Number Formula in number theory, focusing on Lemmas, Proofs, and a significant theorem.

Ring of Polynomials: Coefficients and Multiplication

Covers the ring of polynomials, focusing on coefficients and multiplication.

Sheaf Morphisms and Localization

Explores F-G sheaf morphisms, sheafification, and localization in ring theory.

Algorithms for Big Numbers: Z_n and Orders

Covers algorithms for big numbers, Z_n, and orders in a group, explaining arithmetic operations and cryptographic concepts.

Integrity in Algebra

Explores the properties of commutative rings and integral domains in algebra.

System Equivalence

Explores system equivalence, state-space representation, transfer functions, and Euclidean rings, emphasizing unimodular matrices and their properties.

Classification of Finite Abelian Groups

Covers the classification theorem for finite abelian groups and introduces rings, including zero divisors and domains.

Polynomials: Definition and Operations

Covers polynomials, their operations, division theorem, and provides illustrative examples.

Additivity of Dimension & Height

Explores the additivity of dimension and height in a finitely generated k-algebra domain, showcasing its applications and implications.

Localization in Algebra

Covers the concept of localization in algebra, focusing on making a ring multiplicative closed.

Ring Constructions: Structure Theorems

Explores operations on ideals and structure theorems in commutative rings.

Irreducible Factors and Noetherian Rings

Discusses irreducible factors in rings and the properties of Noetherian rings.

Module Theory: Definitions and Examples

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

The Coordinate Ring of a Linearly Reductive Group

Covers the properties of the coordinate ring of a linearly reductive group.

Ring Theory: Morphisms and Isomorphisms

Covers commutative ring morphisms, subrings, and injective ring morphisms.

Module over a Ring

Explores modules over a ring, injective morphisms, isomorphisms, and canonical groups.

Fermat's Theorem: Sums of Squares

Explores Fermat's Theorem, factorization of integers, properties of Z[i], and Hurwitz quaternions.