Basic notions

• Commutative ring
  Commutative ring
  In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
  
  
• Module (mathematics)
  Module (mathematics)
  In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
  
  
• Ring ideal, maximal ideal
  Maximal ideal
  In mathematics, more specifically in ring theory, a maximal ideal is an ideal which is maximal amongst all proper ideals. In other words, I is a maximal ideal of a ring R if I is an ideal of R, I ≠ R, and whenever J is another ideal containing I as a subset, then either J = I or J = R...
  
  , prime ideal
  Prime ideal
  In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
  
  
• Ring homomorphism
  Ring homomorphism
  In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
  
  
  • Ring monomorphism
  • Ring epimorphism
  • Ring isomorphism
• Zero divisor
  Zero divisor
  In abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
  
  
• Chinese remainder theorem
  Chinese remainder theorem
  The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...
  
  

Classes of rings

• Field (mathematics)
  Field (mathematics)
  In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
  
  
• Algebraic number field
  Algebraic number field
  In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
  
  
• Polynomial ring
  Polynomial ring
  In mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
  
  
• Integral domain
• Boolean algebra (structure)
• Principal ideal domain
  Principal ideal domain
  In abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
  
  
• Euclidean domain
  Euclidean domain
  In mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean algorithm...
  
  
• Unique factorization domain
  Unique factorization domain
  In mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
  
  
• Dedekind domain
  Dedekind domain
  In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
  
  
• Nilpotent
  Nilpotent
  In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....
  
   elements and reduced ring
  Reduced ring
  In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0...
  
  s
• Dual numbers
• Tensor product of fields
  Tensor product of fields
  In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is never itself a field. On the other hand it is often required to 'join' two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K...
  
  
• Tensor product of R-algebras

Constructions with commutative rings

• Quotient ring
  Quotient ring
  In ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
  
  
• Field of fractions
  Field of fractions
  In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
  
  
• Product of rings
  Product of rings
  In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product Πi in I Ri can be turned into a ring by defining the operations coordinatewise, i.e...
  
  
• Annihilator (ring theory)
  Annihilator (ring theory)
  In mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.-Definitions:Let R be a ring, and let M be a left R-module. Choose a nonempty subset S of M...
  
  
• Integral closure

Localization and completion

• Completion (ring theory)
  Completion (ring theory)
  In abstract algebra, a completion is any of several related functors on rings and modules that result in complete topological rings and modules. Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have...
  
  
• Formal power series
  Formal power series
  In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
  
  
• Localization of a ring
  Localization of a ring
  In abstract algebra, localization is a systematic method of adding multiplicative inverses to a ring. Given a ring R and a subset S, one wants to construct some ring R* and ring homomorphism from R to R*, such that the image of S consists of units in R*...
  
  
  • Local ring
    Local ring
    In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
    
    
• Regular local ring
  Regular local ring
  In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...
  
  
• Localization of a module
• Valuation (mathematics)
  • Discrete valuation
  • Discrete valuation ring
    Discrete valuation ring
    In abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...
    
    
• I-adic topology
• Weierstrass preparation theorem
  Weierstrass preparation theorem
  In mathematics, the Weierstrass preparation theorem is a tool for dealing with analytic functions of several complex variables, at a given point P...
  
  

Finiteness properties

• Noetherian ring
  Noetherian ring
  In mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every non-empty set of ideals has a maximal element...
  
  
• Hilbert's basis theorem
  Hilbert's basis theorem
  In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the...
  
  
• Artinian ring
  Artinian ring
  In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
  
  
• Ascending chain condition
  Ascending chain condition
  The ascending chain condition and descending chain condition are finiteness properties satisfied by some algebraic structures, most importantly, ideals in certain commutative rings...
  
   (ACC) and descending chain condition (DCC)

Ideal theory

• Fractional 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...
  
  
• Ideal class group
  Ideal class group
  In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
  
  
• Radical of an ideal
  Radical of an ideal
  In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical...
  
  
• Hilbert's Nullstellensatz
  Hilbert's Nullstellensatz
  Hilbert's Nullstellensatz is a theorem which establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields...
  
  

Homological properties

• Flat module
  Flat module
  In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original...
  
  
• Flat map
  Flat map
  The flat map in differential geometry is a name for the mapping that converts coordinate basis vectors into corresponding coordinate 1-forms.A flat map in ring theory is a homomorphism f from a ring R to a ring S such that S is a flat R-module, where the action of R on S is given by f....
  
  
• Projective module
  Projective module
  In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
  
  
• Injective module
  Injective module
  In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
  
  
• Cohen-Macaulay ring
  Cohen-Macaulay ring
  In mathematics, a Cohen–Macaulay ring is a particular type of commutative ring, possessing some of the algebraic-geometric properties of a nonsingular variety, such as local equidimensionality....
  
  
• Gorenstein ring
  Gorenstein ring
  In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition....
  
  
• Complete intersection ring
• Koszul complex
  Koszul complex
  In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul...
  
  
• Hilbert's syzygy theorem
  Hilbert's syzygy theorem
  In mathematics, Hilbert's syzygy theorem is a result of commutative algebra, first proved by David Hilbert in connection with the syzygy problem of invariant theory. Roughly speaking, starting with relations between polynomial invariants, then relations between the relations, and so on, it...
  
  
• Quillen–Suslin theorem
  Quillen–Suslin theorem
  The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra about the relationship between free modules and projective modules over polynomial rings...
  
  

Dimension theory

• Height (ring theory)
• Depth (ring theory)
  Depth (ring theory)
  In commutative and homological algebra, depth is an important invariant of rings and modules. Although depth can be defined more generally, the most common case considered is the case of modules over a commutative Noetherian local ring. In this case, the depth of a module is related with its...
  
  
• Hilbert polynomial
  Hilbert polynomial
  In commutative algebra, the Hilbert polynomial of a graded commutative algebra or graded module is a polynomial in one variable that measures the rate of growth of the dimensions of its homogeneous components...
  
  
• Regular local ring
  Regular local ring
  In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...
  
  
  • Discrete valuation ring
    Discrete valuation ring
    In abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...
    
    
• Global dimension
  Global dimension
  In ring theory and homological algebra, the global dimension of a ring A denoted gl dim A, is a non-negative integer or infinity which is a homological invariant of the ring. It is defined to be the supremum of the set of projective dimensions of all A-modules...
  
  
• Regular sequence (algebra)
  Regular sequence (algebra)
  In commutative algebra, if R is a commutative ring and M an R-module, a nonzero element r in R is called M-regular if r is not a zerodivisor on M, and M/rM is nonzero...
  
  
• Krull dimension
  Krull dimension
  In commutative algebra, the Krull dimension of a ring R, named after Wolfgang Krull , is the supremum of the number of strict inclusions in a chain of prime ideals. The Krull dimension need not be finite even for a Noetherian ring....
  
  
• Krull's principal ideal theorem
  Krull's principal ideal theorem
  In commutative algebra, Krull's principal ideal theorem, named after Wolfgang Krull , gives a bound on the height of a principal ideal in a Noetherian ring...
  
  

Ring extensions, primary decomposition

• Primary ideal
  Primary ideal
  In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n...
  
  
• Primary decomposition and the Lasker–Noether theorem
  Lasker–Noether theorem
  In mathematics, the Lasker–Noether theorem states that every Noetherian ring is a Lasker ring, which means that every ideal can be written as an intersection of finitely many primary ideals...
  
  
• Noether normalization lemma
  Noether normalization lemma
  In mathematics, the Noether normalization lemma is a result of commutative algebra, introduced in . A simple version states that for any field k, and any finitely generated commutative k-algebra A, there exists a nonnegative integer d and algebraically independent elements y1, y2, ..., yd in Asuch...
  
  
• Going up and going down
  Going up and going down
  In commutative algebra, a branch of mathematics, going up and going down are terms which refer to certain properties of chains of prime ideals in integral extensions....
  
  

Relation with algebraic geometry

• Spectrum of a ring
  Spectrum of a ring
  In abstract algebra and algebraic geometry, the spectrum of a commutative ring R, denoted by Spec, is the set of all proper prime ideals of R...
  
  
• Zariski tangent space
  Zariski tangent space
  In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V...
  
  
• Kähler differential
  Kähler differential
  In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.-Presentation:The idea was introduced by Erich Kähler in the 1930s...
  
  

Computational and algorithmic aspects

• Elimination theory
  Elimination theory
  In commutative algebra and algebraic geometry, elimination theory is the classical name for algorithmic approaches to eliminating between polynomials of several variables....
  
  
• Gröbner basis
  Gröbner basis
  In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R...
  
  
• Buchberger's algorithm
  Buchberger's algorithm
  In computational algebraic geometry and computational commutative algebra, Buchberger's algorithm is a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order. It was invented by Austrian mathematician Bruno Buchberger...
  
  

Related disciplines

• Algebraic number theory
  Algebraic number theory
  Algebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
  
  
• Algebraic geometry
  Algebraic geometry
  Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
  
  
• Ring theory
  Ring theory
  In abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
  
  
• Field theory (mathematics)
  Field theory (mathematics)
  Field theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are well-defined....
  
  
• Differential algebra
  Differential algebra
  In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with a derivation, which is a unary function that is linear and satisfies the Leibniz product law...
  
  
• Homological algebra
  Homological algebra
  Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...