Ring Theory: Definitions and Examples

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Related lectures (32)

Integrity in Algebra

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

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.

Polynomials: Definition and Operations

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

Integral Quadratic Lattices: Properties and Adeles

Covers non-degenerate integral quadratic lattices and adèles, including their properties and definitions.

Chinese Remainder Theorem: Rings and Fields Log in to Mediaspace to watch this video

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

Irreducible Factors and Noetherian Rings Log in to Mediaspace to watch this video

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

Division Rings and Ideals Log in to Mediaspace to watch this video

Explores division rings, integral domains, fields, and ideals in rings, with examples and key theorems.

Ring of Polynomials: Coefficients and Multiplication Log in to Mediaspace to watch this video

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

Module over a Ring Log in to Mediaspace to watch this video

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

Localization in Algebra Log in to Mediaspace to watch this video

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

Dimension Theory of Rings Log in to Mediaspace to watch this video

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

Dedekind Rings: Theory and Applications Log in to Mediaspace to watch this video

Explores Dedekind rings, integral closure, factorization of ideals, and Gauss' Lemma.

Principal Ideal Domains: Structure and Homomorphisms Log in to Mediaspace to watch this video

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

Ring Theory: Sub-rings and Morphisms Log in to Mediaspace to watch this video

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

Congruence Relations in Rings Log in to Mediaspace to watch this video

Explores congruence relations in rings, principal ideals, ring homomorphisms, and the characteristic of rings.

Construction of Quotient Rings Log in to Mediaspace to watch this video

Explores the construction of quotient rings and the properties of well-defined operations within rings.

Ring Theory: Morphisms and Isomorphisms Log in to Mediaspace to watch this video

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

Idempotent Elements and Central Orthogonal Log in to Mediaspace to watch this video

Explores idempotent elements, central orthogonal elements, commutative rings, and prime ideals in non-central rings.

Dimension theory of rings Log in to Mediaspace to watch this video

Covers the dimension theory of rings, including additivity of dimension and height, Krull's Hauptidealsatz, and the height of general complete intersections.

Ideals in Commutative Rings Log in to Mediaspace to watch this video

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

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