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Lecture# Rings and Fields: Principal Ideals and Ring Homomorphisms

Description

This lecture covers the concepts of principal ideals, principal ideal domains, ring homomorphisms, subrings, characteristic of a ring, direct product of rings, commutative rings, zero divisors, integral domains, fields, ideals in a commutative ring, quotient rings, Chinese Remainder Theorem for integers, polynomial rings, Euclidean domains, maximal ideals in polynomial rings, irreducible polynomials, finite fields, and the construction and classification of finite fields.

Official source

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In course

MATH-310: Algebra

This is an introduction to modern algebra: groups, rings and fields.

Instructor

Related concepts (211)

Local ring

In mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or prime. Local algebra is the branch of commutative algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of a ring at a prime ideal.

Artinian ring

In mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided) ideals; that is, there is no infinite descending sequence of ideals. Artinian rings are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are finite-dimensional vector spaces over fields.

Quotient ring

In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. It is a specific example of a quotient, as viewed from the general setting of universal algebra. Starting with a ring R and a two-sided ideal I in R, a new ring, the quotient ring R / I, is constructed, whose elements are the cosets of I in R subject to special + and ⋅ operations.

Principal ideal ring

In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring. If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring.

Fractional ideal

In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity. Let be an integral domain, and let be its field of fractions.

Related lectures (864)

Chinese Remainder Theorem: Rings and FieldsMATH-310: Algebra

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

Finite Fields: Construction and PropertiesMATH-310: Algebra

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

Weyl character formulaMATH-680: Monstrous moonshine

Explores the proof of the Weyl character formula for finite-dimensional representations of semisimple Lie algebras.

Rings and FieldsMATH-310: Algebra

Explores rings, fields, ideals, and their properties in algebraic structures.

Principal Ideal Domains: Structure and HomomorphismsMATH-310: Algebra

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