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...

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

geometrically regular ring is a Noetherian ring over a field that remains a regular ring after any finite extension of the base field. Geometrically regular schemes...

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...

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...

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

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...

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

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...

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...

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...

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

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...

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

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

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 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...

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...

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...

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...

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)...

ring A to be a normal ring. The criterion involves the following two conditions for A: R k : A p {\displaystyle R_{k}:A_{\mathfrak {p}}} is a regular...

varieties and schemes: A point is singular if the local ring at this point is not a regular local ring. Catastrophe theory Defined and undefined Degeneracy...

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...

\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...

"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...

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