Principal Ideal Domains: Structure and Homomorphisms

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Construction of Quotient Rings

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

Finite Fields: Construction and Properties

Explores the construction and properties of finite fields, including irreducible polynomials and the Chinese Remainder Theorem.

Irreducible Factors and Noetherian Rings

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

Division Rings and Ideals

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

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.

Dimension theory of rings

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

Idempotent Elements and Central Orthogonal

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

Polynomials on a Field: Properties and Applications

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

Properties of Euclidean Domains

Covers the properties of Euclidean domains and irreducible elements in polynomial rings.

Division Rings and Ideals

Covers division rings, fields, and ideals in commutative rings with examples in Z and quaternions.