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See all on WikipediaRegular local ring - Wikipedia
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 is equal to its Krull dimension. In symbols, let A be any Noetherian local ring with unique maximal ideal m, and suppose a1, ..., an is a minimal set of generators of m. … See more
There are a number of useful definitions of a regular local ring, one of which is mentioned above. In particular, if $${\displaystyle A}$$ is a Noetherian local ring with maximal ideal $${\displaystyle {\mathfrak {m}}}$$, … See more
The Auslander–Buchsbaum theorem states that every regular local ring is a unique factorization domain.
Every localization, as well as the completion, … See moreIn commutative algebra, a regular ring is a commutative Noetherian ring, such that the localization at every prime ideal is a regular local ring: … See more
1. Every field is a regular local ring. These have (Krull) dimension 0. In fact, the fields are exactly the regular local rings of dimension 0.
2. Any discrete valuation ring is a regular local ring of dimension 1 and the regular local rings of … See moreRegular local rings were originally defined by Wolfgang Krull in 1937, but they first became prominent in the work of Oscar Zariski a few years later, who showed that geometrically, a regular local ring corresponds to a smooth point on an algebraic variety See more
Wikipedia text under CC-BY-SA license Section 10.106 (00NN): Regular local rings—The Stacks project
Regular local rings are defined in Definition 10.60.10. It is not that easy to show that all prime localizations of a regular local ring are regular. In fact, quite a bit of the material developed so far is geared towards a proof of this fact. See Proposition 10.110.5, and trace back the references.
regular local ring in nLab - ncatlab.org
Jun 25, 2024 · A regular local ring R R is a Noetherian commutative unital local ring whose Krull dimension agrees with the minimal number of generators of its maximal ideal I I, equivalently whose Krull dimension equals dimk(I / I2) dim_k (I/I^2) where k = R / I k = R/I is the residue …
The local ring OX;x is regular. There is an open neighborhood U of x which is smooth over k.
There is a range of much-studied properties of local rings. This includes the Cohen–Macaulay property, the Gorenstein property, normality, and regularity. In this book only normality and regularity are dealt with at some length, and one exercise, 13.3, is devoted to the …
Consider a local ring R with unique maximal ideal m. The ideal m is, in particular, an abelian group, and it contains m2 as a normal subgroup, so we can consider the quotient group m=m2, where the group operation is addition of cosets: (m1 + m2) + (m2 + m2) = (m1 + m2) + m2: t is …
A local ring is a Noetherian ring with a single maximal ideal; when we say (R; M) is a local ring we mean that R is a local ring with maximal ideal M. Local rings are unusual, but we can make any Noetherian ring into a local ring using a proccess called localization. A ring R localized at a …
Regular ring (in commutative algebra) - Encyclopedia of …
Jun 6, 2020 · For complete regular local rings, the Cohen structure theorem holds: Such a ring has the form $ R [ [ X _ {1} \dots X _ {n} ] ] $, where $ R $ is a field or a discrete valuation ring. Any module of finite type over a regular local ring has a finite free resolution (see Hilbert …
Regular Local Ring -- from Wolfram MathWorld
Nov 7, 2024 · A regular local ring is a local ring R with maximal ideal m so that m can be generated with exactly d elements where d is the Krull dimension of the ring R. Equivalently, R is regular if the vector space m/m^2 has dimension d.
Section 10.110 (065U): Regular rings and global dimension—The …
By Propositions 10.110.5 and 10.110.1 we see that a Noetherian local ring is a regular local ring if and only if it has finite global dimension. Furthermore, any localization Rp has finite global dimension, see Lemma 10.109.13, and hence is a regular local ring. â–ˇ
What is the relationship between being normal and …
A regular* local ring is a UFD hence integrally closed. In other words, regular implies normal.
Local ring - Encyclopedia of Mathematics
Jun 5, 2020 · If these elements can be chosen in such a way that they generate the maximal ideal $ \mathfrak m $, then $ A $ is called a regular local ring. The regularity of $ A $ is equivalent to the fact that $ \mathop {\rm dim} (\mathfrak m / \mathfrak m ^ {2}) = \mathop {\rm dim} A $.
Section 10.160 (0323): The Cohen structure theorem—The …
The local ring R contains a field. This happens if either \mathbf {Q} \subset R, or pR = 0 where p is the characteristic of R/\mathfrak m. In this case a coefficient ring \Lambda is a field contained in R which maps isomorphically to R/\mathfrak m. The characteristic of R/\mathfrak m is p > 0 but …
1. Prime avoidance and regular local rings are domains ular rings are integral domains. Before continuing however, I need a tronger form of prime avoidance al that isnot necessarily prime. Sup
It follows that the complement of U(k[[X]]) is the set of P anXn al, the ideal generated by X. Hence, k[[X]] is a local ring with unique maximal ideal the ideal generated by X. k[[X]] is called the ring of formal power
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Section 10.18 (07BH): Local rings—The Stacks project
A local ring is a ring with exactly one maximal ideal. If R is a local ring, then the maximal ideal is often denoted mR and the field R/mR is called the residue field of the local ring R.
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Local I.B.M. Rings - The International Brotherhood of Magicians
Find Local Rings Around the World. Please call the I.B.M. Headquarters Office in St. Charles, Missouri, USA (636.724.2400) for additional information regarding possible I.B.M. Rings. Attention, officers: Please contact the I.B.M. Headquarters Office to update your Ring …
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