Local ring - Wikipedia
WEBIn mathematics, more specifically in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on algebraic varieties or manifolds, or of algebraic number fields examined at a particular place, or prime.
Local Ring -- from Wolfram MathWorld
WEBAug 22, 2024 · A local ring is a ring R that contains a single maximal ideal. In this case, the Jacobson radical equals this maximal ideal. One property of a local ring R is that the subset R-m is precisely the set of ring units, where m is the maximal ideal.
WEBp form a complete local ring by definition. They are defined to by Z p = lim ← Z/pnZ. I.e., it is the inverse limit of the sequence ···→ Z/p n→ Z/p −1 →···→ Z/p In other words, Z p is the set of all sequences (a n) of integers a n de-fined modulo pn so that a n + pn−1Z = a n−1. There is a natural ring monomorphism Z ...
Local ring - Encyclopedia of Mathematics
WEBJun 5, 2020 · Local ring. A commutative ring with a unit that has a unique maximal ideal. If $ A $ is a local ring with maximal ideal $ \mathfrak m $, then the quotient ring $ A / \mathfrak m $ is a field, called the residue field of $ A $. Examples of local rings. Any field or valuation ring is local.
Section 10.18 (07BH): Local rings—The Stacks project
WEBA local ring is a ring with exactly one maximal ideal. If $R$ is a local ring, then the maximal ideal is often denoted $\mathfrak m_ R$ and the field $R/\mathfrak m_ R$ is called the residue field of the local ring $R$.
WEBIn commutative algebra, a local ring is defined to be a nonzero ring which has a unique maximal ideal. These local rings form the "local objects" in commutative algebra since for any (commutative) ring R and any prime ideal peR, one can "localize R at p" to get a local ring Rp with unique
local ring in nLab
WEBAug 19, 2024 · An important example of a local ring in algebraic geometry is R = k [ϵ] / ϵ 2 R = k[\epsilon]/\epsilon^2. This ring is known as the ring of dual numbers. Intuitively, we can think of its spectrum as consisting of a closed point and a tangent vector.
WEBThe ring of p-adic integers is its valuation ring, denoted Z p. Proposition 1.7. Z p is the closure of Z inside Q p. In particular Z p is the completion of Z w.r.t. |·| p. Proof. Since Q is dense in Q p and Z p ⊆Q p is open, Z p∩Q is dense in Z p. Note that Z p∩Q = {a b ∈Q |p∤ b}= Z ( ). Thus it suffices to show thatZ is dense in Z ...
WEB1. Local rings Recall a local ring is a ring with a unique maximal ideal. Equivalently, every element outside of this ideal is invertible. Recall that for any ring and prime ideal P R, R P is a local ring with PR P the unique maximal ideal. If you have a homomorphism of local rings, you can put a condition on it that it sort of respects the ...
Local Rings - MathReference
WEBLet R be a local ring. If J is the maximal ideal, also serving as the jacobson radical, then the units of R map onto the units of R/J. Everything in R/J is a unit, hence everything in R-J is a unit. In other words, R minus its units is an ideal.