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 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...

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...

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...

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

a local principal ideal domain, and not a field. R is a valuation ring with a value group isomorphic to the integers under addition. R is a local Dedekind...

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...

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...

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...

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

local rings (see SGA4, Exposé IV, Exercise 13.9), which is equivalent to saying that all the stalks of the structure ring object are local rings when there...

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

endomorphism ring being a local ring. For a semisimple module, the endomorphism ring is a von Neumann regular ring. The endomorphism ring of a nonzero...

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...

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

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...

introduce the "denominators" to a given ring or module. That is, it introduces a new ring/module out of an existing ring/module R, so that it consists of fractions...

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...

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...

dimension two over the reals, and also an Artinian local ring. They are one of the simplest examples of a ring that has nonzero nilpotent elements. Dual numbers...

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...

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...

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

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 algebra, an analytically normal ring is a local ring whose completion is a normal ring, in other words a domain that is integrally closed in its quotient...

In algebra, an analytically irreducible ring is a local ring whose completion has no zero divisors. Geometrically this corresponds to a variety with only...

regular local ring is an integral domain. In fact, a regular local ring is a UFD. The following rings are not integral domains. The zero ring (the ring in...

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...

principal ideal domain, torsion-free modules. Formally, a module M over a ring R is flat if taking the tensor product over R with M preserves exact sequences...