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

multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers...

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...

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

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...

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...

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...

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

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

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...

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 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...

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

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

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...

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...

whose operation is commutative; a commutative ring is a ring whose multiplication is commutative. (Addition in a ring is always commutative.) However, in the...

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

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...

polynomial ring R[X] over a Noetherian ring R is Noetherian. Both facts imply that a finitely generated commutative algebra over a Noetherian ring is again...

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...

over a commutative ring R is a ring homomorphism that is also R-linear. The function f : Z/6Z → Z/6Z defined by f([a]6) = [4a]6 is not a ring homorphism...

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...

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

adjoint" to Hom. The tensor product of two modules A and B over a commutative ring R is defined in exactly the same way as the tensor product of vector...

In algebra, a simplicial commutative ring is a commutative monoid in the category of simplicial abelian groups, or, equivalently, a simplicial object...