Regular local ring - Wikipedia
WEBIn commutative algebra, a regular ring is a commutative Noetherian ring, such that the localization at every prime ideal is a regular local ring: that is, every such localization has the property that the minimal number of generators of …
Regular Ring -- from Wolfram MathWorld
WEBAug 22, 2024 · Regular Ring. A regular ring in the sense of commutative algebra is a commutative unit ring such that all its localizations at prime ideals are regular local rings.
Section 10.106 (00NN): Regular local rings—The Stacks project
WEBRegular local rings. 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.
Regular ring (in commutative algebra) - Encyclopedia of …
WEBJun 6, 2020 · Fields and Dedekind rings are regular rings. If $ A $ is regular, then the ring of polynomials $ A [ X _ {1} \dots X _ {n} ] $ and the ring of formal power series $ A [ [ X _ {1} \dots X _ {n} ] ] $ over $ A $ are also regular.
WEB1. The local ring O X;x is regular. 2. There is an open neighborhood U of x which is smooth over k. Proof: We may assume without loss of generality that X is a ne, say SpecA, where A is the quotient of a polynomial ring B over k by an ideal I. Let Z := SpecB and let m be the maximal ideal of B corresponding to the point x 2X Z. Since the local ...
Regular Local Ring -- from Wolfram MathWorld
WEBAug 22, 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 …
WEBA Noetherian ring $R$ is said to be regular if all the localizations $R_{\mathfrak p}$ at primes are regular local rings. It is enough to require the local rings at maximal ideals to be regular. Note that this is not the same as asking $R$ to have finite global dimension, even assuming $R$ is Noetherian.
von Neumann Regular Ring -- from Wolfram MathWorld
WEBAug 22, 2024 · A von Neumann regular ring is a ring R such that for all a in R, there exists a b in R satisfying a=aba (Jacobson 1989, p. 196). More formally, a ring R is regular in the sense of von Neumann iff the following equivalent conditions hold.
WEBRegular Local Rings. As mentioned before, local rings serve for the study of the local behavior of a global object, such as an affine variety. In particular, notions of local “niceness” can be defined as properties of local rings. There is a range of much-studied properties of local rings.
Regular ring (in the sense of von Neumann)
WEBJun 6, 2020 · Regular rings are a special case of $ \pi $- regular rings, in which, by definition, for every element $ a $ there are an element $ x $ and a positive integer $ n $ such that $ a ^ {n} x a ^ {n} = a ^ {n} $.