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...

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...

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

ring is commutative (that is, its multiplication is a commutative operation) has profound implications on its properties. Commutative algebra, the theory...

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

and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind...

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...

Introduction to Commutative Algebra is a well-known commutative algebra textbook written by Michael Atiyah and Ian G. Macdonald. It is on the list of...

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...

principal ideal domains ⊃ euclidean domains ⊃ fields ⊃ algebraically closed fields An integral domain is a nonzero commutative ring in which the product of any two nonzero...

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...

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

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...

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

ring. Semi-local rings occur for example in algebraic geometry when a (commutative) ring R is localized with respect to the multiplicatively closed subset...

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

Noetherian ring is Noetherian. Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian. (This follows from the two...

Theorem 1. Artin, Michael (2011). Algebra. Prentice Hall. ISBN 978-0-13-241377-0. Bourbaki, N. (1972). Commutative algebra. Paris, Hermann; Reading, Mass...

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...

properties to be the beginning of a cohomology theory of algebraic varieties and of non-commutative rings, there was no clear definition of the higher Kn(X)...

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...

mathematics, dimension theory is the study in terms of commutative algebra of the notion dimension of an algebraic variety (and by extension that of a scheme)....

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

modules over commutative rings have nice properties. The localization of a projective module is a projective module over the localized ring. A projective...

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...

mathematical field of commutative algebra, an ideal I in a commutative ring A is locally nilpotent at a prime ideal p if Ip, the localization of I at p, is a...