Commutative ring information

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

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...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials...

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...

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...

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...

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

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

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

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...

In commutative algebra, the prime spectrum (or simply the spectrum) of a commutative ring R is the set of all prime ideals of R, and is usually denoted...

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...

entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be...

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...

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...

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 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...

(see Polynomial greatest common divisor) and other commutative rings (see § In commutative rings below). The greatest common divisor (GCD) of integers...

algebraic structure that is a vector space over a field or a module over a commutative ring. The collection of all structures of a given type (same operations...

quotient ring R / I is isomorphic to the product of the quotient rings R / In, n = 1, ..., k. An associative algebra A over a commutative ring R is a ring itself...

studies the geometric properties of formal duals of non-commutative algebraic objects such as rings as well as geometric objects derived from them (e.g....

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...