In commutative algebra and algebraic geometry, localization is a formal way to introduce the "denominators" to a given ring or module. That is, it introduces...

Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings....

a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily...

Equivalently, a noncommutative ring is a ring that is not a commutative ring. Noncommutative algebra is the part of ring theory devoted to study of properties...

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist...

Ian G. Macdonald. It deals with elementary concepts of commutative algebra including localization, primary decomposition, integral dependence, Noetherian...

duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g. by gluing along localizations or taking noncommutative...

Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry...

Multiplicative sets are important especially in commutative algebra, where they are used to build localizations of commutative rings. A subset S of a ring R is called...

theorem Roman 2008, p. 115, §4 Ernst Kunz, "Introduction to Commutative algebra and algebraic geometry", Birkhauser 1985, ISBN 0-8176-3065-1 Irving Kaplansky...

generalized sense. A noncommutative algebra is an associative algebra in which the multiplication is not commutative, that is, for which x y {\displaystyle...

In commutative algebra, an integrally closed domain A is an integral domain whose integral closure in its field of fractions is A itself. Spelled out...

algebra that studies commutative local rings and their modules. In practice, a commutative local ring often arises as the result of the localization of...

In commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined...

size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole...

Ring (mathematics) Commutative algebra, Commutative ring Ring theory, Noncommutative ring Algebra over a field Non-associative algebra Relatives to rings:...

the subject. For the items in commutative algebra (the theory of commutative rings), see Glossary of commutative algebra. For ring-theoretic concepts in...

ideal Localization (commutative algebra) Nilpotent – Element in a ring whose some power is 0 Picard group – Mathematical group occurring in algebraic geometry...

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...

In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R {\displaystyle R} is the set of all prime ideals of R {\displaystyle...

the entry-wise addition and multiplication is a two-dimensional commutative algebra over Q {\displaystyle \mathbb {Q} } . However, it is not a field...

facts prove useful in commutative algebra and algebraic geometry: for R ≠ { 0 } {\displaystyle R\neq \lbrace 0\rbrace } commutative, R   /   I {\displaystyle...

last examples are implicitly behind the wide use of localization in commutative algebra and algebraic geometry. For a given ring homomorphism f : A → B...

In commutative algebra, the Auslander–Buchsbaum formula, introduced by Auslander and Buchsbaum (1957, theorem 3.7), states that if R is a commutative Noetherian...

all commutative rings. This category is one of the central objects of study in the subject of commutative algebra. Any ring can be made commutative by...

A commutative algebra over a commutative ring is called a reduced algebra if its underlying ring is reduced. The nilpotent elements of a commutative ring...

relations specialized to the multiplicative semigroup of a commutative ring R. S-units Localization of a ring and a module In the case of rings, the use of...

domain R. Then the localization of R with respect to S is a Z {\displaystyle \mathbb {Z} } -graded ring. If I is an ideal in a commutative ring R, then ⨁...

glossary of commutative algebra. See also list of algebraic geometry topics, glossary of classical algebraic geometry, glossary of algebraic geometry, glossary...

Completion is similar to localization, and together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler...