Commutative algebra

Commutative algebra is a branch of mathematics that studies commutative rings and their ideals. It serves as a foundational area for algebraic geometry, number theory, and various other fields in both pure and applied mathematics. Here are some key concepts and components of commutative algebra: 1. **Rings and Ideals**: A ring is an algebraic structure equipped with two binary operations, typically addition and multiplication, satisfying certain properties.

In mathematics, localization is a technique used to focus on a particular subset of a mathematical structure or to analyze properties of functions, spaces, or objects at a certain point or region. The concept is prevalent in various areas of mathematics, particularly in algebra, topology, and analysis.

The Hasse principle, also known as the local-global principle, is a concept in number theory related to the solvability of equations over the rational numbers (or more generally, over a number field). It states that if a certain equation has solutions in local completions of the field (such as the p-adic numbers for various primes \( p \) and the real numbers), then it should also have a solution in the field itself.

Local analysis is a term that can refer to a variety of analyses depending on the context in which it is used. Generally, it involves examining a specific subset of data or a particular area with a focus on detailed, localized insights. Here are a few contexts where local analysis might apply: 1. **Statistical Analysis**: In statistics, local analysis can refer to examining data within a limited geographic area or a specific subgroup rather than looking at data trends on a larger, more generalized scale.

In algebraic geometry and commutative algebra, a **local ring** is a particular type of ring that has a unique maximal ideal. More formally, if \( R \) is a commutative ring with identity, it is called a local ring if it contains a single maximal ideal \( \mathfrak{m} \). This property leads to a structure that facilitates the study of functions and algebraic entities that are "localized" around a certain point.

A **semi-local ring** is a concept in commutative algebra that generalizes some ideas of local rings. A ring \( R \) is called a semi-local ring if it has a finite number of maximal ideals. This means that the set of maximal ideals of \( R \) is not necessarily just one (as in the case of local rings), but consists of a finite collection of such ideals.

An "acceptable ring" is not a standard term in mathematics, but it could refer to a certain type of algebraic structure known as a "ring" in abstract algebra. In general, a ring is a set equipped with two binary operations that satisfies specific properties.

An **almost ring** is a mathematical structure that generalizes the concept of a ring, with some relaxation of the usual axioms. In particular, an almost ring is defined by a set equipped with two operations (usually called addition and multiplication) that partially satisfy the properties of a ring, but do not necessarily satisfy all the ring axioms. In general, the concept of an almost ring can vary in definition depending on the context or the specific formulation found in various mathematical literature.

An **analytically irreducible ring** is a concept from algebraic geometry and commutative algebra, closely related to the notion of irreducibility in the context of varieties and schemes.

An *analytically normal ring* is a concept that arises in the study of commutative algebra and algebraic geometry, particularly in connection with the behavior of rings of functions. The formal definition typically pertains to rings of functions that arise from algebraic varieties or schemes. A ring \( R \) is said to be **analytically normal** if the following holds: 1. **Integral Closure**: The ring \( R \) is integrally closed in its field of fractions.

An **analytically unramified ring** is a concept from commutative algebra, particularly in the study of local rings and their associated modules. In essence, a local ring is said to be analytically unramified if it behaves well with respect to analytic geometry over its residue field.

An Arf ring is a specific type of commutative ring in the field of algebra, particularly in the study of algebraic topology and homotopy theory. It is named after the mathematician Michael Arf, who contributed significantly to the theory of forms and associated structures.

The Artin approximation theorem is a result in algebraic geometry and number theory that deals with the behavior of power series and their solutions in a local ring setting. Specifically, it is concerned with the approximation of solutions to polynomial equations.

In algebra, specifically in the theory of rings and modules, an *Artinian ideal* typically refers to an ideal in a ring that satisfies the descending chain condition (DCC). This means that any descending chain of ideals within an Artinian ideal eventually stabilizes; that is, there are no infinite descending sequences. More generally, a ring is called an *Artinian ring* if it satisfies the descending chain condition for ideals.

The Ascending Chain Condition (ACC) is a property related to partially ordered sets (posets) and certain algebraic structures in mathematics, particularly in order theory and abstract algebra. **Definition:** A partially ordered set satisfies the Ascending Chain Condition if every ascending chain of elements eventually stabilizes.

An **atomic domain** is a concept in the field of mathematics, specifically in the area of ring theory, which is a branch of abstract algebra. A domain is a specific type of ring that has certain properties, and an atomic domain is a further classification of such a ring. In general, a **domain** (often referred to as an integral domain) is a commutative ring with no zero divisors and where the multiplication operation is closed.

The Auslander–Buchsbaum formula is a significant result in commutative algebra and homological algebra that relates the projective dimension of a module to its depth and the dimension of the ring over which the module is defined. Specifically, it provides a way to compute the projective dimension of a finitely generated module over a Noetherian ring.

The Auslander–Buchsbaum theorem is a fundamental result in the field of commutative algebra, specifically in the study of modules over local rings and their projective dimensions. It provides a connection between the dimensions of modules and their resolutions.

The Bass number, denoted as \( b(G) \), is an important concept in the study of graph theory and algebraic topology. It measures the number of "independent" cycles in a graph or topological space. Specifically, in the context of algebraic topology, it can relate to the concept of Betti numbers and the structure of a simplicial complex.

The Bass–Quillen conjecture is a conjecture in the field of algebraic K-theory, specifically concerning finitely generated infinite projective modules over a commutative ring. It was formulated by mathematicians Hyman Bass and Daniel Quillen in the 1970s.

A **Buchsbaum ring** is a type of commutative ring that has certain desirable properties, particularly in the context of algebraic geometry and commutative algebra. It is named after the mathematician David Buchsbaum.

A Bézout domain is a specific type of integral domain in abstract algebra that possesses a particular property related to the linear combinations of its elements.

A catenary ring is a type of structural element that takes the form of a curve known as a catenary, which is the shape that a hanging flexible chain or rope assumes under its own weight when supported at its ends. In architectural and engineering contexts, catenary rings are used to create stable and efficient structures, often in the design of arches, bridges, and roof systems. The mathematical equation for a catenary curve is typically expressed in terms of hyperbolic functions.

In algebra, the concept of **change of rings** involves the study of a ring homomorphism and how it allows us to transfer structures and properties from one ring to another. This is particularly relevant in areas like algebraic geometry, representation theory, and commutative algebra.

Cluster algebras are a class of commutative algebras that were introduced by mathematician Laurent F. Robbin in 2001. They have a rich structure and have connections to various areas of mathematics, including combinatorics, representation theory, and algebraic geometry. ### Key Features of Cluster Algebras 1. **Clusters and Variables**: A cluster algebra is constructed using sets of variables called "clusters." Each cluster consists of a finite number of variables.

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.

A **finitely generated algebra** is a specific type of algebraic structure that is built from a vector space over a field (often denoted \( K \)) by introducing a multiplication operation. The key aspect of a finitely generated algebra is that it can be constructed using a finite number of generators. More formally, let \( A \) be a vector space over a field \( K \).

In the context of algebraic geometry and commutative algebra, a **fitting ideal** is a specific type of ideal associated with a module over a ring. It captures information about the relations between elements of the module. For a finitely generated module \(M\) over a Noetherian ring \(R\), the Fitting ideals provide a way of understanding the structure of \(M\) in terms of its generators and relations.

In differential geometry and related fields, a **formally smooth map** generally refers to a type of map that behaves smoothly at a certain level, even if it may not be globally smooth in the traditional sense across its entire domain. The concept is often discussed in the context of algebraic geometry and singularity theory. To provide a clearer understanding: 1. **Smooth Maps**: A smooth map is typically a function between differentiable manifolds that is infinitely differentiable.

The term "G-ring" can refer to several different concepts depending on the context, such as mathematics, chemistry, or other specialized fields. However, it is most commonly known in the context of algebra, specifically in ring theory. In mathematics, a **G-ring** typically refers to a **generalized ring**, which is a structure that generalizes the concept of a ring by relaxing some of the usual requirements.

In mathematics, a **GCD domain** (which stands for **Greatest Common Divisor domain**) is a type of integral domain that possesses certain properties regarding the divisibility of its elements. Specifically, an integral domain \( D \) is classified as a GCD domain if it satisfies the following conditions: 1. **Integral Domain:** \( D \) must be an integral domain (meaning it is a commutative ring with no zero divisors and has a multiplicative identity).

A geometrically regular ring is a concept that arises in algebraic geometry and commutative algebra. Specifically, it relates to geometric properties of the spectrum of a ring, particularly in regard to its points and their corresponding field extensions.

A glossary of commutative algebra is a collection of terms and definitions that are commonly used in the field of commutative algebra, which is a branch of mathematics that studies commutative rings, their ideals, and modules over those rings. Here are some key terms and concepts typically found in such a glossary: 1. **Ring**: A set equipped with two binary operations (addition and multiplication) that satisfy certain properties (associativity, distributivity, etc.).

The phrases "going up" and "going down" can refer to various contexts depending on the subject matter. Here are a few interpretations: 1. **General Meaning**: - "Going up" often denotes an increase or upward movement, such as in prices, stock values, or in physical elevation (like climbing a hill). - "Going down" typically indicates a decrease or downward movement, such as falling prices, declining values, or descending physically.

A **Gorenstein ring** is a type of commutative ring that has particularly nice homological properties. More formally, a Noetherian ring \( R \) is called Gorenstein if it satisfies the following equivalent conditions: 1. **Dualizing Complex**: The singularity category of \( R \) has a dualizing complex which is concentrated in non-negative degrees, and the homological dimension of the ring is finite.

A Hahn series is a formal power series that arises in the context of ordered groups and valuation theory. Specifically, it is used to describe a way to represent elements of certain fields, particularly in relation to ordered abelian groups.

Hausdorff completion is a mathematical process used to construct a complete metric space from a given metric space that may not be complete. The idea is to extend the space in such a way that all Cauchy sequences converge within the new space. ### Overview of the Process: 1. **Metric Spaces and Completeness**: A metric space is a set equipped with a distance function (metric) that defines how far apart the points are.

A Henselian ring is a type of commutative ring that satisfies a certain property related to the completeness of its valuation. More specifically, a ring \( R \) is called Henselian if it is equipped with a valuation \( v \) such that certain conditions hold, particularly that the ring is complete with respect to this valuation, and that certain polynomial equations behave like they do in a complete local field.

Hilbert's Basis Theorem is a fundamental result in algebra, particularly in the theory of rings and ideals. It states that if \( R \) is a Noetherian ring (meaning that every ideal in \( R \) is finitely generated), then any ideal in the polynomial ring \( R[x] \) (the ring of polynomials in one variable \( x \) with coefficients in \( R \)) is also finitely generated.

Hilbert's Syzygy Theorem is a fundamental result in the field of commutative algebra and algebraic geometry that concerns the relationships among generators of modules over polynomial rings. It provides a deeper insight into the structuring of polynomial ideals and their resolutions. In simple terms, the theorem addresses the projective resolutions of finitely generated modules over a polynomial ring.

The concepts of **Hilbert series** and **Hilbert polynomial** arise primarily in algebraic geometry and commutative algebra, particularly in the study of graded algebras and projective varieties. ### Hilbert Series The **Hilbert series** of a graded algebra (or a graded module) is a generating function that encodes the dimensions of its graded components.

The Hilbert–Samuel function is an important concept in commutative algebra and algebraic geometry, particularly in the study of the structure of space defined by ideals in rings and the geometry of schemes. It provides a way to measure the growth of the dimensions of the graded components of the quotient of a Noetherian ring by an ideal.

Hironaka decomposition is a concept in the context of algebraic geometry and singularity theory, specifically related to the resolution of singularities. The term is often associated with the work of Heisuke Hironaka, who is well-known for his theorem on the resolution of singularities in higher-dimensional spaces.

Hodge algebra is a concept in mathematics that arises in the study of Hodge theory, which is a field connecting algebraic topology, differential geometry, and algebraic geometry. Hodge theory is centered on the decomposition of differential forms on a smooth manifold and the study of their topological and geometric properties. More formally, a Hodge algebra typically refers to a certain type of graded algebra that arises in the context of Hodge theory, particularly when considering cohomology and the Hodge decomposition theorem.

Homological conjectures in commutative algebra refer to a collection of important and influential conjectures that relate to the behavior of modules over rings, particularly regarding their homological properties. These conjectures often involve investigating the relationships between various homological dimensions of modules (such as projective dimension, injective dimension, and global dimension) and their implications for ring theory and algebraic geometry.

The \(I\)-adic topology is a concept from algebraic number theory and algebraic geometry that generalizes the notion of topology in the context of ideals in rings, specifically in relation to \(p\)-adic numbers.

The term "ideal norm" can have different meanings depending on the context. Here are a couple of interpretations based on various fields: 1. **Mathematics/Statistics**: In the context of mathematics, particularly in functional analysis and linear algebra, an "ideal norm" could refer to the notion of a norm that satisfies certain properties or conditions ideal for a given space.

"Ideal reduction" can refer to different concepts depending on the context in which it is used. Here are a few interpretations based on various fields: 1. **Mathematics / Algebra**: In the context of algebraic structures, "ideal reduction" might refer to the process of simplifying algebraic expressions or problems using ideals in ring theory. An ideal is a special subset of a ring that can be used to create quotient rings, facilitating the study of various properties of the ring.

Ideal theory is a concept primarily associated with political philosophy and ethics, particularly in discussions surrounding justice, fairness, and the principles that should govern a well-ordered society. It can refer to the formulation of theoretical frameworks or principles that define what an ideal society should look like and how individuals within it should behave.

An **integral domain** is a specific type of algebraic structure in the field of abstract algebra. It is defined as a non-zero commutative ring with certain properties.

An **integrally closed domain** is a type of integral domain in which every element that is integral over the domain is already an element of the domain itself. To understand this concept, let's break it down: 1. **Integral Domain**: An integral domain is a commutative ring with no zero divisors and a multiplicative identity (usually denoted as 1). It also has the property that it is non-trivial (the ring is not the zero ring).

"Introduction to Commutative Algebra" is a well-known textbook written by David Eisenbud, which provides a comprehensive overview of the field of commutative algebra. It serves as an accessible entry point for students and researchers delving into the subject. Commutative algebra is a branch of algebra that studies commutative rings and their ideals, focusing on properties and structures that arise from these algebraic constructs.

In the context of ring theory, an irreducible ring is typically referred to as a ring that cannot be factored into "simpler" rings in a specific way.

In the context of abstract algebra, particularly in ring theory, an **irrelevant ideal** is typically discussed in relation to the properties of ideals in polynomial rings or local rings. While the term "irrelevant ideal" may not be universally defined across all mathematics literature, it's most commonly associated with certain ideals in the study of algebraic geometry and commutative algebra.

The J-2 ring, also known simply as a J-ring, refers to a particular type of ring in the study of algebraic structures in mathematics. Specifically, a J-2 ring is a ring where a certain condition related to Jacobson radical and nilpotent elements holds.

J-multiplicity is a concept that appears in the context of mathematical logic and model theory, particularly in the study of structures and their properties. It is often associated with the analysis of certain functions or relations over structures, and can be used to investigate how complex a particular model or theory is.

In the context of commutative algebra, a Jacobson ring is a ring that satisfies certain properties related to its prime ideals and maximal ideals. Specifically, a ring \( R \) is called a **Jacobson ring** if the intersection of all maximal ideals of \( R \) is equal to the nilradical of \( R \).

The Koszul–Tate resolution is a construction in algebraic geometry and homological algebra used to study certain algebraic structures, particularly those that involve differential forms or algebraic relations. It is named after Jean-Pierre Serre and William Tate, who contributed to the understanding of such resolutions. In simple terms, the Koszul-Tate resolution provides a way to resolve algebraic objects, such as modules or complexes associated with algebraic varieties, using tools from homological algebra.

Krull's Principal Ideal Theorem is a significant result in commutative algebra that connects the concept of prime ideals to the structure of a ring. Specifically, it provides conditions under which a principal ideal generated by an element in a Noetherian ring intersects non-trivially with a prime ideal. The theorem states the following: Let \( R \) be a Noetherian ring, and let \( P \) be a prime ideal of \( R \).

A **Krull ring** is a specific type of commutative ring that has certain ideal-theoretic properties. Named after Wolfgang Krull, these rings are important in algebraic geometry and commutative algebra due to their connection to the concept of dimension and the behavior of their prime ideals.

Commutative algebra is a branch of mathematics that studies commutative rings and their ideals, as well as their applications to algebraic geometry and other areas of mathematics. Here is a list of various topics commonly covered in commutative algebra: 1. **Basic Concepts:** - Rings and ring homomorphisms - Ideals and quotient rings - Prime ideals and maximal ideals - Integral domains and fields 2.

The local criterion for flatness is a condition in algebraic geometry and commutative algebra that helps determine when a morphism (or ring homomorphism) is flat. Flatness is an important property that relates to how properties of rings (or varieties) behave under base change.

The term "local parameter" can have different meanings depending on the context in which it is used. Here are a few possible interpretations: 1. **In Mathematics**: A local parameter often refers to a variable that is used within a limited scope or specific region of a mathematical function or model. For example, in topology, local parameters can describe local properties of spaces or functions.

In the context of algebra, particularly in ring theory and module theory, a module (or a ring) is said to be **locally nilpotent** if every finitely generated submodule (or ideal) has a nilpotent element. More formally, an element \( x \) in a ring (or module) is nilpotent if there exists some positive integer \( n \) such that \( x^n = 0 \).

In the context of ring theory, a **minimal prime ideal** is a prime ideal \( P \) in a commutative ring \( R \) such that there are no other prime ideals contained within \( P \) except for \( P \) itself. In other words, \( P \) is a minimal element in the set of prime ideals of the ring with respect to inclusion.

The Monomial Conjecture, proposed by mathematician G. G. Szegő in 1939 and later expanded upon, concerns the topology and combinatorial mathematics of polytopes and their connection to the algebraic properties of certain spaces. It posits that certain types of generating functions, particularly those related to monomials in polynomial rings, can be understood through the topology of specific polytopes.

A Mori domain is a concept in the field of algebraic geometry, particularly in the study of algebraic varieties and their properties. It is a type of algebraic structure that arises in the context of Mori theory, which is concerned with the classification of algebraic varieties and the birational geometry of these varieties. In more specific terms, a Mori domain is typically a normal, irreducible, and properly graded algebraic domain that satisfies certain conditions related to the Mori program.

The Mori–Nagata theorem is a result in algebraic geometry, particularly concerning the structure of algebraic varieties and their properties under certain conditions. Named after Shigeo Mori and Masayuki Nagata, the theorem deals with the existence of a specific type of morphism called a "rational map" between varieties.

In the context of mathematics, particularly in the fields of algebra and number theory, a **multiplicatively closed set** is a subset of a given set that is closed under the operation of multiplication. This means that if you take any two elements from this set and multiply them together, the result will also be an element of the set. Formally, let \( S \) be a set.

In algebraic geometry and commutative algebra, a **multiplier ideal** is a conceptual tool used to study the properties of singularities of algebraic varieties and to generalize notions of regularity and divisor theory. Multiplier ideals arise in the context of *Cohen-Macaulay* rings and provide a way to handle sheaf-theoretic aspects of the geometry of varieties.

A Nagata ring is a special type of ring in commutative algebra. More specifically, it is a class of rings that are defined in the context of properties related to integral closure and integral extensions.

The nilradical of a ring is an important concept in ring theory, a branch of abstract algebra. Specifically, the nilradical of a ring \( R \) is defined as the set of all nilpotent elements in \( R \). An element \( x \) of \( R \) is called nilpotent if there exists some positive integer \( n \) such that \( x^n = 0 \).

In mathematics, a Novikov ring is a specific type of algebraic structure that arises in the context of algebraic topology and homological algebra, particularly in the study of loop homology and more generally in the theory of algebraic spaces that involve formal power series.

A **parafactorial local ring** is a specific type of local ring that possesses unique factorization properties in a manner that extends the concept of unique factorization in integers or principal ideal domains (PIDs). To understand a parafactorial local ring, let's start breaking down the key components involved: 1. **Local Ring**: A local ring is a ring that has a unique maximal ideal.

In the context of mathematics, particularly in abstract algebra, a **perfect ideal** is a concept that can arise in the theory of rings. However, the term "perfect ideal" is not standard and could be used in various contexts with slightly different meanings depending on the specific area of study.

In the context of ring theory, a branch of abstract algebra, a **primal ideal** typically refers to a specific type of ideal in a commutative ring. However, the term can sometimes lead to confusion, as its definition can vary slightly depending on the context or the source.

Primary decomposition is a concept in the field of algebra, particularly in commutative algebra and algebraic geometry, that deals with the structure of ideals in a ring, specifically Noetherian rings. The primary decomposition theorem provides a way to break down an ideal into a union of 'primary' ideals.

In ring theory, a branch of abstract algebra, a **primary ideal** is a specific type of ideal that has certain properties related to the concept of prime ideals.

In the context of abstract algebra, specifically in ring theory, a principal ideal is a specific type of ideal in a ring that can be generated by a single element. Formally, let \( R \) be a ring and let \( a \) be an element of \( R \).

A **Principal Ideal Domain (PID)** is a special type of integral domain in the field of abstract algebra. Here are some key characteristics of a PID: 1. **Integral Domain**: A PID is an integral domain, which means it is a commutative ring with no zero divisors and has a multiplicative identity (usually denoted as 1). 2. **Principal Ideals**: In a PID, every ideal is a principal ideal.

A **principal ideal ring** (PIR) is a type of ring in which every ideal is a principal ideal. This means that for any ideal \( I \) in the ring \( R \), there exists an element \( r \in R \) such that \( I = (r) = \{ r \cdot a : a \in R \} \). In other words, each ideal can be generated by a single element.

A Prüfer domain is a type of integral domain that generalizes the notion of a Dedekind domain. It is defined as an integral domain \( D \) in which every finite non-zero torsion-free ideal is a projective module. This property is very similar to that of Dedekind domains, which states that every non-zero fractional ideal is a projective \( D \)-module.