Commutative algebra | Ideals (ring theory)
In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element which is to say the set of all elements less than or equal to in The remainder of this article addresses the ring-theoretic concept. (Wikipedia).
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Abstract Algebra | Ideals of quotients of PIDs
We prove that every ideal of a quotient of a principal ideal domain is also principal. Notice that the new space may not be an integral domain, so it is sometimes called a principal ring. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http:
From playlist Abstract Algebra
[ANT04] Counting ideal classes
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Abstract Algebra | k[x] is a PID
We prove that every ideal of the polynomial ring k[x], where k is a field, is principal. That is, k[x] is a principal ideal domain. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http://www.michael-penn.net Randolph College Math: http://www
From playlist Abstract Algebra
Rings and modules 4 Unique factorization
This lecture is part of an online course on rings and modules. We discuss unique factorization in rings, showing the implications (Integers) implies (Euclidean domain) implies (Principal ideal domain) implies (Unique factorization domain). We give a few examples to illustrate these implic
From playlist Rings and modules
Abstract Algebra | Principal Ideals of a Ring
We define the notion of a principal ideal of a ring and give some examples. We also prove that all ideals of the integers are principal ideals, that is, the integers form a principal ideal domain (PID). http://www.michael-penn.net https://www.researchgate.net/profile/Michael_Penn5 http://
From playlist Abstract Algebra
Abstract Algebra | Introduction to Principal Ideal Domains (PIDs)
After introducing the notion of a principal ideal domain (pid), we give some examples, and prove some simple results. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1 Personal Website: http://www.michael-penn.net Randolph College Math: http://www.randolphcolle
From playlist Abstract Algebra
Lecture 8. PIDs and Euclidean domains
From playlist Abstract Algebra 2
Schemes 39: Divisors and Dedekind domains
This lecture is part of an online algebraic geometry course on schemes, based on chapter II of "Algebraic geometry" by Hartshorne. In this lecture we describe Weil and Cartier divisors for Dedekind domains, showing that they correspond to the two classical ways of defining the class group
From playlist Algebraic geometry II: Schemes
Abstract Algebra | Every PID is a UFD.
We prove the classical result in commutative algebra that every principal ideal domain is in fact a unique factorization domain. Along the way, we introduce the ascending chain condition and the notion of a Noetherian ring. Please Subscribe: https://www.youtube.com/michaelpennmath?sub_co
From playlist Abstract Algebra
CTNT 2018 - "Arithmetic Statistics" (Lecture 2) by Álvaro Lozano-Robledo
This is lecture 2 of a mini-course on "Arithmetic Statistics", taught by Álvaro Lozano-Robledo, during CTNT 2018, the Connecticut Summer School in Number Theory. For more information about CTNT and other resources and notes, see https://ctnt-summer.math.uconn.edu/
From playlist CTNT 2018 - "Arithmetic Statistics" by Álvaro Lozano-Robledo
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