Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Dedekind Rings and Fractional Ideals

In courses (2)

MATH-482: Number theory I.a - Algebraic number theory

Algebraic number theory is the study of the properties of solutions of polynomial equations with integral coefficients; Starting with concrete problems, we then introduce more general notions like algebraic number fields, algebraic integers, units, ideal c ...

MATH-317: Algebra V - Galois theory

Galois theory lies at the interface of Field Theory and Group Theory. It aims to describe the algebraic symmetries of fields. We will focus on Galois theory for finite field extensions and some applications. ...

Description

This lecture covers the concept of Dedekind rings and fractional ideals, focusing on Proposition 3.6 which states the existence of a fractional ideal p^-1 such that p.p = A in a Dedekind ring. It further explores the properties of fractional ideals as A-modules and their relationship with maximal ideals. The lecture delves into the integrally closed nature of A, the factorization of fractional ideals into prime ideals, and the extension to fractional ideals. Additionally, it discusses the isomorphism of fractional ideals and their structure as a commutative group generated by Spec(A).

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Instructors (3)

Philippe Michel

Ph. Michel's main research interest lie in the field of analytic number theory and range over a variety of techniques and methods which include: arithmetic geometry, exponential sums, sieve methods, automorphic forms and allied representations, L-functions and more recently ergodic theory. Ph. Michel is a former student of ENS Cachan and obtained his PhD in Universté Paris XI in 1995 under the guidance of E. Fouvry. From 1995 to 1998 he was maître de conférence at Universté Paris XI and full professor at Université Montpellier II until 2008 then when he joined EPFL. Ph. Michel was awarded the Peccot-Vimont prize, has been member of the Institut Universitaire de France and wa invited speaker at the 2006 International Congress of Mathematician.

Thomas Gerber

Donna Testerman

Official source

Ontological neighbourhood

Mathematics

Algebra: Abstract algebra

Related concepts (32)

Ideal (ring theory)

In mathematics, and more specifically in ring theory, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers preserves evenness, and multiplying an even number by any integer (even or odd) results in an even number; these closure and absorption properties are the defining properties of an ideal.

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.

Maximal ideal

In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all proper ideals. In other words, I is a maximal ideal of a ring R if there are no other ideals contained between I and R. Maximal ideals are important because the quotients of rings by maximal ideals are simple rings, and in the special case of unital commutative rings they are also fields.

Prime ideal

In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal. Primitive ideals are prime, and prime ideals are both primary and semiprime. An ideal P of a commutative ring R is prime if it has the following two properties: If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P, P is not the whole ring R.

Free module

In mathematics, a free module is a module that has a basis, that is, a generating set consisting of linearly independent elements. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commutative case), then there exist non-free modules. Given any set S and ring R, there is a free R-module with basis S, which is called the free module on S or module of formal R-linear combinations of the elements of S. A free abelian group is precisely a free module over the ring Z of integers.

Related lectures (37)

Ideals and Representations Log in to Mediaspace to watch this video

Covers ideals, representations, modules, and maximal ideals in associative algebras.

Dedekind Rings: Integral Extensions and Noetherian Rings Log in to Mediaspace to watch this video

Explores Dedekind rings, integral extensions, and noetherian rings in algebraic structures.

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.

Separable Extensions: Dedekind Rings Log in to Mediaspace to watch this video

Explores separable extensions and Dedekind rings, focusing on coefficients and prime ideals.

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