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...

Neumann regular rings should not be confused with the unrelated regular rings and regular local rings of commutative algebra. An element a of a ring is called...

ring over k. Broadly speaking, regular local rings are somewhat similar to polynomial rings. Regular local rings are UFD's. Discrete valuation rings are...

assumptions, a local ring is Cohen–Macaulay exactly when it is a finitely generated free module over a regular local subring. Cohen–Macaulay rings play a central...

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

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

Cohen–Macaulay ring. A regular local ring is an example of a Cohen–Macaulay ring. It is a theorem of Serre that R is a regular local ring if and only if it has...

In commutative algebra, a regular sequence is a sequence of elements of a commutative ring which are as independent as possible, in a precise sense. This...

A 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...

intersection rings ⊃ regular local rings A Gorenstein ring is a commutative Noetherian ring such that each localization at a prime ideal is a Gorenstein local ring...

commutative algebra, a G-ring or Grothendieck ring is a Noetherian ring such that the map of any of its local rings to the completion is regular (defined below)...

inclusions. Universally catenary rings ⊃ Cohen–Macaulay rings ⊃ Gorenstein rings ⊃ complete intersection rings ⊃ regular local rings Suppose that A is a Noetherian...

ring Local ring Noetherian and artinian rings Ordered ring Poisson ring Reduced ring Regular ring Ring of periods SBI ring Valuation ring and discrete...

explicit example is the ring of integers Z, a Euclidean domain. All regular local rings are integrally closed as well. A ring whose localizations at all...

theory) Integral closure Completion (ring theory) Formal power series Localization of a ring Local ring Regular local ring Localization of a module Valuation...

concepts of homological algebra. Let R be a Noetherian, commutative, regular local ring and let P and Q be prime ideals of R. Serre defined the intersection...

is called a Cohen–Macaulay ring if its dimension is equal to its depth. A regular local ring is an example of such a ring. A Noetherian integral domain...

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...

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...

  Absolutely regular is an alternative term for geometrically regular. 6.  An absolutely simple point is one with a geometrically regular local ring. acceptable...

ideal is a finite extension of a regular local ring. The Weierstrass preparation theorem can be used to show that the ring of convergent power series over...

a regular local ring, contains a non-empty open subset, a J-1 ring is a ring such that the set of regular points is an open subset, and a J-2 ring is...

the formal power series ring R[[X]] over R is not a UFD. The Auslander–Buchsbaum theorem states that every regular local ring is a UFD. Z [ e 2 π i n...

Neumann regular ring Quasi-Frobenius ring Hereditary ring, Semihereditary ring Local ring, Semi-local ring Discrete valuation ring Regular local ring Cohen–Macaulay...

finitely generated free module over a polynomial subalgebra or a regular local ring. Such decompositions are named after Heisuke Hironaka, who used this...

of commutative rings may be defined as the rings such that two dimensions are equal; for example, a regular ring is a commutative ring such that the homological...

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...

positivity conjecture that if R {\displaystyle R} is a commutative regular local ring, and P , Q {\displaystyle P,Q} are prime ideals of R {\displaystyle...

commutative algebra, the Auslander–Buchsbaum theorem states that regular local rings are unique factorization domains. The theorem was first proved by...

\Lambda =\mathbb {Z} _{p}[[\Gamma ]]} . This is a 2-dimensional, regular local ring, and this makes it possible to describe modules over it. From this...