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 - 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 mR and the field R/mR is called the residue field of the local ring R.
Local Ring -- from Wolfram MathWorld
WEBAug 22, 2024 · A local ring is a ring that contains a single maximal ideal. In this case, the Jacobson radical equals this maximal ideal. One property of a local ring is that the subset is precisely the set of ring units, where is the maximal ideal.
Section 10.153 (04GE): Henselian local rings—The Stacks project
WEB10.153 Henselian local rings. In this section we discuss a bit the notion of a henselian local ring. Let (R,m, κ) be a local ring. For a ∈ R we denote a¯¯¯ the image of a in κ. For a polynomial f ∈ R[T] we often denote f¯¯¯ the image of f in κ[T].
WEB1. Local Rings. lements of A is a left ideal. (The reason for the name will become apparent later when we discuss the process of \localization" which a. ways results in local rings.) In principle we would have to distinguish between \left" local and \right" local, but fortunately the following facts make. that dis. all .
WEBA 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.
Comm. Algebra - Local Rings - Stanford University
WEBLocal Rings. We call a ring R R local if R R has exactly one maximal ideal M M. In this case, we call A/M A / M the residue field of R R. A ring with only finitely many maximal ideals is called semi-local. Example: Any field F F is local and F F is its own residue field.
Abstract Algebra and Discrete Mathematics, Local Rings
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.
Local Rings - MathReference
WEBYou have probably seen local rings defined this way: R is local if it has a unique maximal ideal. That's fine if R is commutative. Here is a more general definition. The ring R is local if it has a unique maximal left ideal.
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