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

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

mathematics, a noncommutative ring is a ring whose multiplication is not commutative; that is, there exist a and b in the ring such that ab and ba are different...

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

nonzero ring R in which every nonzero element is a unit (that is, R× = R ∖ {0}) is called a division ring (or a skew-field). A commutative division ring is...

commutative algebra, and algebraic geometry. In ring theory, many classes of rings, such as unique factorization domains, regular rings, group rings,...

associative commutative algebra. Replacing the field of scalars by a commutative ring leads to the more general notion of an algebra over a ring. Algebras...

examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major...

beyond number rings to the setting of polynomial rings and other commutative rings by David Hilbert and especially Emmy Noether. Given a ring R, a left ideal...

mathematics, an associative algebra A over a commutative ring (often a field) K is a ring A together with a ring homomorphism from K into the center of A...

mathematics, a ringed space is a family of (commutative) rings parametrized by open subsets of a topological space together with ring homomorphisms that...

right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics...

space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a module also generalizes the notion of an abelian...

is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers...

Ring is a commutative ring. The action of a monoid (= commutative ring) R on an object (= ring) A of Ring is an R-algebra. The category of rings has a number...

algebra homomorphism between unital associative algebras over a commutative ring R is a ring homomorphism that is also R-linear. The function f : Z/6Z → Z/6Z...

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

faithful module. Throughout this section, let R {\displaystyle R} be a commutative ring and M {\displaystyle M} a finitely generated R {\displaystyle R} -module...

In algebra, the nilradical of a commutative ring is the ideal consisting of the nilpotent elements: N R = { f ∈ R ∣ f m = 0  for some  m ∈ Z > 0 } . {\displaystyle...

a b–1 ≠ b–1 a. A commutative division ring is a field. Wedderburn's little theorem asserts that all finite division rings are commutative and therefore finite...

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...

In algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous...

together they are among the most basic tools in analysing commutative rings. Complete commutative rings have a simpler structure than general ones, and Hensel's...

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...

and prime ideals are both primary and semiprime. An ideal P of a commutative ring R is prime if it has the following two properties: If a and b are two...

In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many...

variables. Likewise, the polynomial ring may be regarded as a free commutative algebra. For R a commutative ring, the free (associative, unital) algebra...

G Ring may refer to: Rings of Saturn § G Ring, a planetary ring system around Saturn. G-ring or Grothendieck ring, a type of commutative ring in algebra...

associative", just as "noncommutative" means "not necessarily commutative" for noncommutative rings. An algebra is unital or unitary if it has an identity element...

In commutative algebra, a Gorenstein local ring is a commutative Noetherian local ring R with finite injective dimension as an R-module. There are many...