Commutative algebra is the branch of algebra that studies commutative rings, the ideals inside them and the modules built over them, forming the algebraic foundation that both algebraic number theory and algebraic geometry are built on. This description is adapted from Wikipedia contributors under CC BY-SA 4.0; changes were made. https://creativecommons.org/licenses/by-sa/4.0/
Facts
Central QuestionWhich properties of a commutative ring, above all the structure of its ideals, determine whether elements factor uniquely, and how the ascending chain condition Emmy Noether formalized separates rings with tractable structure from those without it. 1 Key DebateHow far Wolfgang Krull's introduction of localization and completion could carry the subject toward a fully geometric picture of a ring. Krull's principal ideal theorem is widely considered the field's single most important foundational result, and the abstract, ring-based approach Hilbert and Noether pioneered displaced the older, more computational methods of classical invariant theory, a methodological shift that took decades to complete. 1 Classification
Pure or Applied Connections
Associated With
Includes
Source Wikipedia: Abraham Seidenberg
Source Artin-Rees Lemma (Wikipedia)
Source Bass-Quillen Conjecture, Wikipedia
Source Elementary Symmetric Polynomial (Wikipedia)
Source Wikipedia: Emanuel Lasker
Source Going up and going down (Wikipedia)
Source Homological Conjectures in Commutative Algebra (Wikipedia)
Source Wikipedia: Irvin S. Cohen
Source Krull Dimension (Wikipedia)
Source Krull-Akizuki theorem (Wikipedia)
Source Krull's principal ideal theorem (Wikipedia)
Source Primary decomposition (Wikipedia)
Source Monomial Conjecture, Wikipedia
Source Noether normalization lemma, Wikipedia
Source Serre's Multiplicity Conjectures (Wikipedia)
Source Spectrum of a Ring (Wikipedia)
Source Wolfgang Krull (Wikipedia)
Sources
1. Commutative Algebra (Wikipedia)
WikipediaIntroduction
the branch of algebra that studies commutative rings, their ideals, and modules over such rings
Overview
Commutative algebra is essentially the study of the rings occurring in algebraic number theory and algebraic geometry
History (Noether)
recast many earlier results in terms of an ascending chain condition, now known as the Noetherian condition
History (Krull)
The main figure responsible for the birth of commutative algebra as a mature subject was Wolfgang Krull, who introduced the fundamental notions of localization and completion of a ring
History (Krull's theorem)
Krull's principal ideal theorem is widely considered the single most important foundational theorem in commutative algebra
View the Source Monomial Conjecture, Wikipedia
Includes: Monomial Conjecture, Lead sentenceQuote, Includes: Monomial Conjecture, Lead sentence
In commutative algebra, a field of mathematics, the monomial conjecture of Melvin Hochster says the following: Let A be a Noetheri
Homological Conjectures in Commutative Algebra (Wikipedia)
Includes: Homological Conjectures in Commutative Algebra, Lead sentenceQuote, Includes: Homological Conjectures in Commutative Algebra, Lead sentence
ectures have been a focus of research activity in commutative algebra since the early 1960s.
Going up and going down (Wikipedia)
Includes: Going-Up and Going-Down Theorems, Lead sentenceQuote, Includes: Going-Up and Going-Down Theorems, Lead sentence
In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of
Elementary Symmetric Polynomial (Wikipedia)
Includes: Elementary Symmetric Polynomial, Lead sentenceQuote, Includes: Elementary Symmetric Polynomial, Lead sentence
In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for
Krull Dimension (Wikipedia)
Includes: Krull Dimension, Lead sentenceQuote, Includes: Krull Dimension, Lead sentence
In commutative algebra, the Krull dimension of a commutative ring R, named after Wolfgang Krull, is the supremum of the lengths of
Spectrum of a Ring (Wikipedia)
Includes: Spectrum of a Ring, Lead sentenceQuote, Includes: Spectrum of a Ring, Lead sentence
In mathematics, and more specifically in commutative algebra and algebraic geometry, the prime spectrum (or simply the spectrum) o
Krull-Akizuki theorem (Wikipedia)
Serre's Multiplicity Conjectures (Wikipedia)
Wikimedia FoundationIncludes: Serre's Multiplicity Conjectures, Lead sentenceQuote, Includes: Serre's Multiplicity Conjectures, Lead sentence
after Jean-Pierre Serre, are certain problems in commutative algebra, motivated by the needs of algebraic geometry.
Noether normalization lemma, Wikipedia
Includes: Noether Normalization Lemma, Lead sentenceQuote, Includes: Noether Normalization Lemma, Lead sentence
as a theorem rather than a lemma) is a result of commutative algebra, introduced by Emmy Noether in 1926.
Krull's principal ideal theorem (Wikipedia)
Wolfgang Krull (Wikipedia)
Includes: Wolfgang Krull, Lead paragraph [in-branch]Quote, Includes: Wolfgang Krull, Lead paragraph [in-branch]
commutative algebra
Wikipedia: Irvin S. Cohen
Includes: Irvin S. Cohen, LeadQuote, Includes: Irvin S. Cohen, Lead
who worked on local rings
Wikipedia: Abraham Seidenberg
Includes: Abraham Seidenberg, LeadQuote, Includes: Abraham Seidenberg, Lead
Seidenberg was known for his research in commutative algebra
Wikipedia: Emanuel Lasker
Includes: Emanuel Lasker, LeadQuote, Includes: Emanuel Lasker, Lead
known for his contributions to commutative algebra
Bass-Quillen Conjecture, Wikipedia
Includes: Bass-Quillen Conjecture, Lead sectionQuote, Includes: Bass-Quillen Conjecture, Lead section
the Bass-Quillen conjecture relates vector bundles over a regular Noetherian ring A and over the polynomial ring A [ t 1 , …
Artin-Rees Lemma (Wikipedia)
Wikimedia FoundationIncludes: Artin-Rees Lemma, Lead sectionQuote, Includes: Artin-Rees Lemma, Lead section
mathematics, the Artin-Rees lemma is a basic result about modules over a Noetherian ring, along with results such as the Hilbert basis
Primary decomposition (Wikipedia)
Includes: Lasker-Noether Theorem, Lead sectionQuote, Includes: Lasker-Noether Theorem, Lead section
mathematics, the Lasker-Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be decomposed
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