A Cohen ring is a concept from algebraic geometry and commutative algebra, primarily related to the study of algebraic varieties and their functions. Specifically, it often arises in the context of the reduction of schemes and local rings. A Cohen ring is associated with a geometric object such as a local ring of a scheme, particularly in the study of the structure of complete local rings.
The Cohen structure theorem, named after Paul Cohen, is a result in set theory and mathematical logic that addresses the structure of certain kinds of sets of reals or more generally, in the context of set-theoretic topology. The theorem is particularly important in the study of forcing and independence results in mathematics. In simple terms, the Cohen structure theorem describes the nature of a model of set theory obtained by adding generic reals through a forcing construction known as Cohen forcing.
A Cohen-Macaulay ring is a type of commutative ring with specific geometric and algebraic properties, often used in algebraic geometry and commutative algebra.
A complete intersection is a concept from algebraic geometry that refers to a type of geometric object defined by the intersection of multiple subvarieties in a projective or affine space. Specifically, a variety \( X \) is called a complete intersection if it can be defined as the common zero set of a certain number of homogeneous or non-homogeneous polynomial equations, and if the number of equations is equal to the codimension of the variety.
In algebraic geometry and commutative algebra, a **complete intersection ring** is associated with a particular kind of algebraic variety, namely those that can be defined as the common zeros of a certain number of polynomials in a polynomial ring. To provide a clearer understanding, let’s go through some definitions step by step. 1. **Algebraic Variety**: An algebraic variety is a geometric object that is the solution set of a system of polynomial equations.
The concept of completion of a ring is a fundamental idea in algebra, especially in the context of commutative algebra, number theory, and algebraic geometry. Completing a ring typically involves creating a new ring that captures the "local" behavior of the original ring with respect to a given ideal.
In ring theory, a branch of abstract algebra, a **conductor** is a specific concept used to describe a relationship between two rings, particularly in the context of commutative rings with unity.
The term "congruence ideal" is primarily used in the context of algebra, particularly in the study of rings and ideals in ring theory. Although it's not as commonly referenced as some other concepts, the idea generally relates to how certain elements of a ring or algebraic structure can be used to define relationships and equivalences among elements. In the context of a ring \( R \), a congruence relation is an equivalence relation that is compatible with the ring operations.
A connected ring typically refers to a type of network topology used in computer science and telecommunications. In a connected ring topology, each device (or node) in the network is connected to exactly two other devices, forming a circular shape or "ring." This means that data can be transmitted in one direction (or sometimes both directions) around the ring.
Constructible topology is a concept in the field of mathematical logic and set theory, particularly in the context of model theory and the foundations of mathematics. It is used to study the properties of sets and their relationships with various mathematical structures. In the constructible universe, denoted as \( L \), sets are built in a hierarchical manner using definable sets based on certain criteria.
The concept of deviation in the context of local rings can refer to different things depending on the specific mathematical setting. However, in algebraic geometry and commutative algebra, the term "deviation" is often related to the concept of "dualizing complexes", "canonical modules", or even to certain homological dimensions relative to local rings.
Differential calculus over commutative algebras is a branch of mathematics that generalizes the concepts of differentiation and integration from classical calculus to the context of commutative algebras, which are algebraic structures that satisfy certain properties, notably that multiplication is commutative.
Differential graded algebra (DGA) is a mathematical structure that combines concepts from algebra and topology, particularly in the context of homological algebra and algebraic topology. A DGA consists of a graded algebra equipped with a differential that satisfies certain properties. Here’s a more detailed breakdown of the components and properties: ### Components of a Differential Graded Algebra 1.
A discrete valuation ring (DVR) is a specific type of integral domain that has useful properties in algebraic geometry and number theory. Here are the key characteristics of a discrete valuation ring: 1. **Integral Domain**: A DVR is an integral domain, which means it is a commutative ring with no zero divisors and has a multiplicative identity (1 ≠ 0).
The term "divided domain" can refer to several concepts depending on the context in which it is used. Here are a few interpretations: 1. **Mathematics and Set Theory**: In mathematics, particularly in set theory and analysis, a divided domain may refer to a partitioned set where a domain is split into distinct subdomains or subsets. Each subset can be analyzed independently, often to simplify complex problems or to study properties that hold for each subset.
In the context of commutative algebra and algebraic geometry, the dualizing module is an important concept that arises in the study of schemes and their cohomological properties. ### Definition Given a Noetherian ring \( R \), the dualizing module is an \( R \)-module \( \mathcal{D} \) that serves as a kind of "dual" object to the module of differentials.
The Eakin–Nagata theorem is a result in the field of functional analysis and specifically concerns the relationship between certain ideals in the context of Banach spaces and their duals. This theorem is particularly relevant in the study of dual spaces and the structure of various function spaces.
A Euclidean domain is a type of integral domain (a non-zero commutative ring with no zero divisors) that satisfies a certain property similar to the division algorithm in the integers.
An "excellent ring" typically refers to a concept in the field of algebra, specifically in the area of commutative algebra and algebraic geometry. In these contexts, a ring is called **excellent** if it satisfies certain desirable properties that make it behave nicely with respect to various algebraic operations.
Finite algebra refers to algebraic structures that are defined on a finite set. These structures can include groups, rings, fields, and other algebraic systems, all of which have a finite number of elements. Here are a few key points regarding finite algebra: 1. **Finite Groups**: A group is a set equipped with a binary operation that satisfies four properties: closure, associativity, the presence of an identity element, and the existence of inverses.