specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the...

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 mathematics, a semi-local ring is a ring for which R/J(R) is a semisimple ring, where J(R) is the Jacobson radical of R. (Lam 2001, p. §20)(Mikhalev...

a ring Simplicial commutative ring Special types of rings: Boolean ring Dedekind ring Differential ring Exponential ring Finite ring Lie ring Local ring...

In algebraic geometry, a local ring A is said to be unibranch if the reduced ring Ared (obtained by quotienting A by its nilradical) is an integral domain...

deviations of a local ring R are certain invariants εi(R) that measure how far the ring is from being regular. The deviations εn of a local ring R with residue...

In algebraic geometry, a Noetherian local ring R is called parafactorial if it has depth at least 2 and the Picard group Pic(Spec(R) − m) of its spectrum...

mathematics, a Cohen–Macaulay ring is a commutative ring with some of the algebro-geometric properties of a smooth variety, such as local equidimensionality. Under...

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

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...

catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings A local complete intersection ring is a Noetherian...

In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were introduced by Azumaya (1951), who named them...

mathematics, specifically abstract algebra, an Artinian ring (sometimes Artin ring) is a ring that satisfies the descending chain condition on (one-sided)...

geometrically regular local ring. acceptable ring Acceptable rings are generalizations of excellent rings, with the conditions about regular rings in the definition...

case considered is the case of modules over a commutative Noetherian local ring. In this case, the depth of a module is related with its projective dimension...

following equivalent conditions: R is a local ring, a principal ideal domain, and not a field. R is a valuation ring with a value group isomorphic to the...

A principal ideal domain that is not a field has Krull dimension 1. A local ring has Krull dimension 0 if and only if every element of its maximal ideal...

integers. Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings (group rings, division...

equicharacteristic Noetherian local ring is a ring of formal power series over a field. (Equicharacteristic means that the local ring and its residue field have...

states that a projective module over a local ring is free; where a not-necessarily-commutative ring is called local if for each element x, either x or 1...

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

terminology, points with regular local rings were called simple points, and points with geometrically regular local rings were called absolutely simple points...

particular, every valuation ring is a local ring. The valuation rings of a field are the maximal elements of the set of the local subrings in the field partially...

its parts. A ring that is a semisimple module over itself is known as an Artinian semisimple ring. Some important rings, such as group rings of finite groups...

Noetherian rings need not be well-behaved: for example, a normal Noetherian local ring need not be analytically normal. The class of excellent rings was defined...

In ring theory, a branch of mathematics, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is...

"localization of a ring", "local ring", "regular ring". An affine algebraic variety corresponds to a prime ideal in a polynomial ring, and the points of...

Look up ring in Wiktionary, the free dictionary. (The) Ring(s) may refer to: Ring (jewellery), a round band, usually made of metal, worn as ornamental...

of the ring; by the Krull intersection theorem, this is the case for any commutative Noetherian ring which is an integral domain or a local ring. There...

or invertible element of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists...