Dimension | Commutative algebra

Krull dimension

In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of all chains of prime ideals. The Krull dimension need not be finite even for a Noetherian ring. More generally the Krull dimension can be defined for modules over possibly non-commutative rings as the deviation of the poset of submodules. The Krull dimension was introduced to provide an algebraic definition of the dimension of an algebraic variety: the dimension of the affine variety defined by an ideal I in a polynomial ring R is the Krull dimension of R/I. A field k has Krull dimension 0; more generally, k[x1, ..., xn] has Krull dimension n. A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal is nilpotent. There are several other ways that have been used to define the dimension of a ring. Most of them coincide with the Krull dimension for Noetherian rings, but can differ for non-Noetherian rings. (Wikipedia).

Video thumbnail

Dimensions (1 of 3: The Traditional Definition - Directions)

More resources available at www.misterwootube.com

From playlist Exploring Mathematics: Fractals

Video thumbnail

What is a dimension? In 3D...and 2D... and 1D

1D - it's the new 3D! Tweet it - http://bit.ly/mP3FFo Facebook it - http://on.fb.me/qtTraR minutephysics is now on Google+ - http://bit.ly/qzEwc6 And facebook - http://facebook.com/minutephysics Minute Physics provides an energetic and entertaining view of old and new problems

From playlist MinutePhysics

Video thumbnail

There is no "Fourth" dimension

Just because there are four dimensions doesn't mean there's a "fourth dimension" 4D rubik's cube: http://www.superliminal.com/cube/cube.htm minutephysics is now on Google+ - http://bit.ly/qzEwc6 And facebook - http://facebook.com/minutephysics And twitter - @minutephysics Minute Physic

From playlist Relativity

Video thumbnail

Commutative algebra 57: Krull versus Hilbert

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We continue the previous video by showing that the Krull dimension of a Noetherian local ring is at most the dimension defined

From playlist Commutative algebra

Video thumbnail

Commutative algebra 55: Dimension of local rings

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We give 4 definitions of the dimension of a Noetherian local ring: Brouwer-Menger-Urysohn dimension, Krull dimension, degree o

From playlist Commutative algebra

Video thumbnail

Commutative algebra 59: Krull's principal ideal theorem

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We give some applications of the theorems we proved about the dimension of local rings. We first show that the dimension of a

From playlist Commutative algebra

Video thumbnail

Commutative algebra 58: System of parameters versus Krull

This lecture is part of an online course on commutative algebra, following the book "Commutative algebra with a view toward algebraic geometry" by David Eisenbud. We show that the smallest size of a system of parameters of a Noetherian local ring is at most the Krull dimension. The proof

From playlist Commutative algebra

Video thumbnail

Dimensions Chapter 5

Chapter 5 of the Dimensions series. See http://www.dimensions-math.org for more information. Press the 'CC' button for subtitles.

From playlist Dimensions

Video thumbnail

Emmy Noether in Erlangen and Göttingen by Ravi Rao

DATES: Monday 29 Aug, 2016 - Tuesday 30 Aug, 2016 VENUE: Madhava Lecture Hall, ICTS Bangalore Emmy Noether (1882­-1935) is well known for her famous contributions to abstract algebra and theoretical physics. Noether’s mathematical work has been divided into three ”epochs”. In the first (

From playlist The Legacy of Emmy Noether

Video thumbnail

Origin and Development of Valuation Theory by Sudesh Khanduja

DATES: Monday 29 Aug, 2016 - Tuesday 30 Aug, 2016 VENUE: Madhava Lecture Hall, ICTS Bangalore Emmy Noether (1882­-1935) is well known for her famous contributions to abstract algebra and theoretical physics. Noether’s mathematical work has been divided into three ”epochs”. In the first (

From playlist The Legacy of Emmy Noether

Video thumbnail

Seminar on Applied Geometry and Algebra (SIAM SAGA): Sonja Petrović

Date: Tuesday, January 12, 2021 at 11:00am EST (5:00pm CET) Speaker: Sonja Petrović, Illinois Institute of Technology Title: Random Monomial Ideals Abstract: Joint work with Jesus A. De Loera, Lily Silverstein, Despina Stasi, Dane Wilburne. Inspired by the study of random graphs and sim

From playlist Seminar on Applied Geometry and Algebra (SIAM SAGA)

Video thumbnail

Dimensions Chapter 6

Chapter 6 of the Dimensions series. See http://www.dimensions-math.org for more information. Press the 'CC' button for subtitles.

From playlist Dimensions

Video thumbnail

The Curse of Oak Island: TWO HUGE FINDS IN ONE DAY (Season 8) | History

While continuing to excavate more sections of the stone pathway, the team stumbles upon two significant finds - a cuff button and an old fire grate, in this clip from Season 8, "Off the Railing." #OakIsland Watch all new episodes of The Curse of Oak Island, Tuesdays at 9/8c, and stay up

From playlist Forged in Fire: Season 8 | New Episodes Return Wednesday, March 24 at 9/8c | History

Video thumbnail

The Curse of Oak Island: MYSTERIOUS METAL Reveals Secret Hatch Location (Season 9)

Gary Drayton and the team uncover compelling clues in their search for the hatch location, in this clip from Season 9, "The Root Cause." Watch the new episodes of The Curse of Oak Island, Tuesdays at 9/8c, and stay up to date on all of your favorite The HISTORY Channel shows at history.co

From playlist The Curse of Oak Island: Season 9 | History

Video thumbnail

From cluster categories to scattering diagrams (Lecture 1) by Bernhard Keller

PROGRAM :SCHOOL ON CLUSTER ALGEBRAS ORGANIZERS :Ashish Gupta and Ashish K Srivastava DATE :08 December 2018 to 22 December 2018 VENUE :Madhava Lecture Hall, ICTS Bangalore In 2000, S. Fomin and A. Zelevinsky introduced Cluster Algebras as abstractions of a combinatoro-algebra

From playlist School on Cluster Algebras 2018

Related pages

Dimension of an algebraic variety | Integral domain | Minimal prime ideal | Zero ring | Annihilator (ring theory) | Deviation of a poset | Regular local ring | Maximal ideal | Commutative algebra | Artinian ring | Cohen–Macaulay ring | Depth (ring theory) | Principal ideal domain | Polynomial ring | Direct product | Noetherian | Affine variety | Dedekind domain | Dimension theory (algebra) | Normal cone | Von Neumann regular ring | Field (mathematics) | Gelfand–Kirillov dimension | Algebraic geometry | Noetherian ring | Codimension | Reduced ring | Associated graded ring | Discrete valuation ring | Homological conjectures in commutative algebra | Krull's principal ideal theorem | Scheme (mathematics) | Galois connection | Local ring | Noether normalization lemma | Spectrum of a ring | Unique factorization domain | Catenary ring | Module (mathematics) | Commutative ring