 List of commutative algebra topics

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent examples of commutative rings include polynomial rings, rings of algebraic integers, including the ordinary integers , and padic integers.
Research fields
Active research areas
 Serre's multiplicity conjectures
 Homological conjectures
Basic notions
 Commutative ring
 Module (mathematics)
 Ring ideal, maximal ideal, prime ideal
 Ring homomorphism
 Ring monomorphism
 Ring epimorphism
 Ring isomorphism
 Zero divisor
 Chinese remainder theorem
Classes of rings
 Field (mathematics)
 Algebraic number field
 Polynomial ring
 Integral domain
 Boolean algebra (structure)
 Principal ideal domain
 Euclidean domain
 Unique factorization domain
 Dedekind domain
 Nilpotent elements and reduced rings
 Dual numbers
 Tensor product of fields
 Tensor product of Ralgebras
Constructions with commutative rings
 Quotient ring
 Field of fractions
 Product of rings
 Annihilator (ring theory)
 Integral closure
Localization and completion
 Completion (ring theory)
 Formal power series
 Localization of a ring
 Regular local ring
 Localization of a module
 Valuation (mathematics)
 Iadic topology
 Weierstrass preparation theorem
Finiteness properties
 Noetherian ring
 Hilbert's basis theorem
 Artinian ring
 Ascending chain condition (ACC) and descending chain condition (DCC)
Ideal theory
Main article: Ideal theoryHomological properties
 Flat module
 Flat map
 Projective module
 Injective module
 CohenMacaulay ring
 Gorenstein ring
 Complete intersection ring
 Koszul complex
 Hilbert's syzygy theorem
 Quillen–Suslin theorem
Dimension theory
Main article: Dimension theory (algebra) Height (ring theory)
 Depth (ring theory)
 Hilbert polynomial
 Regular local ring
 Global dimension
 Regular sequence (algebra)
 Krull dimension
 Krull's principal ideal theorem
Ring extensions, primary decomposition
 Primary ideal
 Primary decomposition and the Lasker–Noether theorem
 Noether normalization lemma
 Going up and going down
Relation with algebraic geometry
Computational and algorithmic aspects
Active research areas
 Serre's multiplicity conjectures
 homological conjectures
Related disciplines
 Algebraic number theory
 Algebraic geometry
 Ring theory
 Field theory (mathematics)
 Differential algebra
 Homological algebra
Outlines General reference · Culture and the arts · Geography and places · Health and fitness · History and events · Mathematics and logic · Natural and physical sciences · People and self · Philosophy and thinking · Religion and belief systems · Society and social sciences · Technology and applied sciencesCategories: Mathematicsrelated lists
 Commutative algebra
 Outlines
Wikimedia Foundation. 2010.
Look at other dictionaries:
List of abstract algebra topics — Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this … Wikipedia
List of homological algebra topics — This is a list of homological algebra topics, by Wikipedia page. Basic techniques Cokernel Exact sequence Chain complex Differential module Five lemma Short five lemma Snake lemma Nine lemma Extension (algebra) Central extension Splitting lemma… … Wikipedia
List of group theory topics — Contents 1 Structures and operations 2 Basic properties of groups 2.1 Group homomorphisms 3 Basic types of groups … Wikipedia
List of algebraic geometry topics — This is a list of algebraic geometry topics, by Wikipedia page. Contents 1 Classical topics in projective geometry 2 Algebraic curves 3 Algebraic surfaces 4 … Wikipedia
List of algebraic topology topics — This is a list of algebraic topology topics, by Wikipedia page. See also: topology glossary List of topology topics List of general topology topics List of geometric topology topics Publications in topology Topological property Contents 1… … Wikipedia
List of category theory topics — This is a list of category theory topics, by Wikipedia page. Specific categories *Category of sets **Concrete category *Category of vector spaces **Category of graded vector spaces *Category of finite dimensional Hilbert spaces *Category of sets… … Wikipedia
List of mathematics articles (L) — NOTOC L L (complexity) L BFGS L² cohomology L function L game L notation L system L theory L Analyse des Infiniment Petits pour l Intelligence des Lignes Courbes L Hôpital s rule L(R) La Géométrie Labeled graph Labelled enumeration theorem Lack… … Wikipedia
List of algebraic structures — In universal algebra, a branch of pure mathematics, an algebraic structure is a variety or quasivariety. Abstract algebra is primarily the study of algebraic structures and their properties. Some axiomatic formal systems that are neither… … Wikipedia
Algebra — This article is about the branch of mathematics. For other uses, see Algebra (disambiguation). Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from… … Wikipedia
List of mathematical topics in quantum theory — This is a list of mathematical topics in quantum theory, by Wikipedia page. See also list of functional analysis topics, list of Lie group topics, list of quantum mechanical systems with analytical solutions. Contents 1 Mathematical formulation… … Wikipedia