Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols to solve equations and understand relationships between quantities. At its core, algebra involves the use of letters (often referred to as variables) to represent numbers or values in mathematical expressions and equations. Key concepts in algebra include: 1. **Variables**: Symbols (usually letters) that represent unknown values (e.g., \( x \), \( y \)).
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Abstract algebra is a branch of mathematics that studies algebraic structures, which are sets equipped with operations that satisfy certain axioms. The main algebraic structures studied in abstract algebra include: 1. **Groups**: A group is a set equipped with a single binary operation that satisfies four properties: closure, associativity, the existence of an identity element, and the existence of inverses. Groups can be finite or infinite and are foundational in many areas of mathematics.
In the context of Wikipedia and similar collaborative projects, "stubs" refer to articles that are incomplete and provide insufficient information on a topic. They are essentially minimal entries that may be just a couple of paragraphs long and need more content to adequately cover the subject matter.
In the context of Wikipedia and other collaborative platforms, "stubs" refer to short articles that provide only a limited amount of information on a particular topic. An "Algebraic geometry stub" specifically pertains to a page related to algebraic geometry that is incomplete, lacking in detail, or requires expansion. Algebraic geometry is a field of mathematics that studies the solutions of systems of algebraic equations and their geometric properties.
In the context of category theory, a "stub" typically refers to a brief or incomplete article or entry about a concept, topic, or theorem within the broader field of category theory. It often indicates that the information provided is minimal and that the article requires expansion or additional detail to fully cover the topic. This can include definitions, examples, applications, and important results related to category theory. Category theory itself is a branch of mathematics that deals with abstract structures and the relationships between them.
In the context of Wikipedia, "Commutative algebra stubs" refers to short articles or entries related to the field of commutative algebra that need expansion or additional detail. A "stub" is generally a brief piece of writing that provides minimal information about a topic, often requiring more comprehensive content to adequately cover the subject. Commutative algebra itself is a branch of mathematics that studies commutative rings and their ideals, with applications in algebraic geometry, number theory, and other areas.
In the context of Wikipedia and other collaborative online encyclopedias, a "stub" refers to a very short article or entry that provides minimal information on a given topic but is intended to be expanded over time. Group theory stubs, therefore, are entries related to group theory—an area of abstract algebra that studies algebraic structures known as groups—that lack sufficient detail, thoroughness, or breadth.
Abel's irreducibility theorem is a result in algebra that concerns the irreducibility of certain polynomials over the field of rational numbers (or more generally, over certain fields).
The Alperin–Brauer–Gorenstein theorem is a result in group theory regarding the structure of finite groups. Specifically, it deals with the existence of groups that have certain properties with respect to their normal subgroups and the actions of their Sylow subgroups.
The Andreotti–Grauert theorem is a result in complex geometry and several complex variables. It addresses the properties of complex spaces and certain types of submanifolds known as complex manifolds. The theorem specifically relates to the existence of holomorphic (complex-analytic) functions on compact complex manifolds.
Aperiodic semigroups are a concept from algebra, specifically within the study of semigroup theory. A semigroup is a set equipped with an associative binary operation. To understand aperiodicity in this context, it's essential to delve into some definitions associated with semigroups.
An Arf semigroup is a specific type of algebraic structure studied in the context of commutative algebra and algebraic geometry, especially in the theory of integral closures of rings and in the classification of singularities.
An **arithmetic ring**, commonly referred to as an **arithmetic system** or simply a **ring**, is a fundamental algebraic structure in the field of abstract algebra. Specifically, a ring is a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication.
The Artin-Zorn theorem is a result in the field of set theory and is often discussed in the context of ordered sets and Zorn's lemma. It specifically deals with the existence of maximal elements in certain partially ordered sets under certain conditions.
Auslander algebra is a concept in representation theory and homological algebra, primarily associated with the study of finitely generated modules over rings. The topic is named after the mathematician Maurice Auslander, who made significant contributions to both representation theory and commutative algebra. At its core, the Auslander algebra of a module category is constructed from the derived category of finitely generated modules over a particular ring.
An automorphism of a Lie algebra is a specific type of isomorphism that is defined within the context of Lie algebras. To be more precise, consider a Lie algebra \( \mathfrak{g} \) over a field (commonly the field of real or complex numbers).
The Baer–Suzuki theorem is a result in group theory that deals with the structure of groups, specifically p-groups, and the conditions under which certain types of normal subgroups can be constructed. The theorem is part of a broader study in the representation of groups and the interplay between their normal subgroups and group actions.
A Brandt semigroup is a specific type of algebraic structure that arises in the context of semigroups, which are sets equipped with an associative binary operation. More formally, a Brandt semigroup is defined as follows: A Brandt semigroup is a semigroup of the form \( B_{n}(G) \) for some positive integer \( n \) and some group \( G \).
The Brauer–Fowler theorem is a result in the field of group theory, more specifically in the study of linear representations of finite groups. It deals with the structure of certain finite groups and their representations over fields with certain characteristics.
The Brauer–Nesbitt theorem is a result in the theory of representations of finite groups, specifically pertaining to the representation theory of the symmetric group. The theorem characterizes the irreducible representations of a symmetric group \( S_n \) in terms of their behavior with respect to certain arithmetic functions.
The Brauer–Suzuki theorem is a result in group theory, specifically in the area of representation theory and the theory of finite groups. Named after mathematicians Richard Brauer and Michio Suzuki, the theorem provides important conditions for the existence of certain types of groups and their representations. One of the most prominent statements of the Brauer–Suzuki theorem pertains to the structure of finite groups, characterizing when a certain kind of simple group can be singly generated by an element of specific order.
The Brauer–Suzuki–Wall theorem is a result in group theory, specifically in the area of representation theory. The theorem deals with the characterization of certain types of groups, known as \( p \)-groups, and their representation over fields of characteristic \( p \).
A CAT(k) group is a type of geometric group that arises in the study of metric spaces and their large-scale geometric properties. The term "CAT(k)" comes from the work of mathematician Mikhail Gromov and relates to CAT(0) spaces, which are simply connected spaces that have non-positive curvature in a very generalized sense. In this context, a **CAT(k)** space is a geodesic metric space that satisfies a condition related to triangles.
CEP stands for "Centralizer-Infinitely Generated Abelian Part." In the context of group theory, the CEP subgroup of a group is a specific subset that captures certain properties of the group's structure. The concept of the CEP subgroup is often related to the study of groups in terms of their centralizers, which are subgroups formed by elements that commute with a given subset of the group.
The Carnot group is a specific type of mathematical structure found in the field of differential geometry and geometric analysis, often studied within the context of sub-Riemannian geometry and metric geometry. In particular, Carnot groups are a class of nilpotent Lie groups that can be understood in terms of their underlying algebraic structures.
The Cartan–Brauer–Hua theorem is a result in the field of representation theory and the theory of algebraic groups, particularly regarding representations of certain classes of algebras. It mainly deals with the representation theory of semisimple algebras and is associated with the work of mathematicians Henri Cartan, Richard Brauer, and Shiing-Shen Chern, who made significant contributions to the understanding of group representations and the structure of algebraic objects.
The Cartan–Dieudonné theorem is a result in differential geometry and linear algebra that characterizes elements of a projective space using linear combinations of certain vectors. Specifically, it is often described in the context of the geometry of vector spaces and the projective spaces constructed from them.
A **Chinese monoid** refers to a specific algebraic structure that arises in the study of formal language theory and algebra. The term may not be widely referenced in mainstream mathematical literature outside of specific contexts, but it may relate to the concept of monoids in general. A **monoid** is defined as a set equipped with an associative binary operation and an identity element.
A "clean ring" is a term that can refer to different concepts depending on the context in which it is used. However, it is not a widely recognized term in any specific discipline.
A **Clifford semigroup** is a specific type of algebraic structure in the study of semigroups, particularly within the field of algebra. A semigroup is a set equipped with an associative binary operation. Specifically, a Clifford semigroup is defined as a commutative semigroup in which every element is idempotent.
A **cocompact group action** refers to a specific type of action of a group on a topological space, particularly in the context of topological groups and geometric topology. In broad terms, if a group \( G \) acts on a topological space \( X \), we say that the action is **cocompact** if the quotient space \( X/G \) is compact.
In the context of group theory, a complemented group is a specific type of mathematical structure, particularly within the study of finite groups. A group \( G \) is said to be **complemented** if, for every subgroup \( H \) of \( G \), there exists a subgroup \( K \) of \( G \) such that \( K \) is a complement of \( H \).
The term "complete field" can refer to different concepts depending on the context. Here are a few possible interpretations: 1. **Mathematics (Field Theory)**: In algebra, a "field" is a set equipped with two operations that generalize the arithmetic of the rational numbers. A "complete field" might refer to a field that is complete with respect to a particular norm or metric.
In the context of topology and algebraic topology, the term "component theorem" can refer to several different theorems concerning the structure of topological spaces, graphs, or abstract algebraic structures like groups or rings. However, without a specific area of mathematics in mind, it’s challenging to pin down exactly which "component theorem" you are referring to.
In mathematics, particularly in the study of field theory, a **composite field** is formed by taking the combination (or extension) of two or more fields.
Congruence-permutable algebras are a class of algebras studied in universal algebra and related fields. An algebraic structure is generally described by a set along with a collection of operations and relations defined on that set. The concept of congruences in algebra refers to certain equivalence relations that respect the operations of the algebra.
In group theory, a subgroup \( H \) of a group \( G \) is called **conjugacy-closed** if, for every element \( h \) in \( H \) and every element \( g \) in \( G \), the conjugate \( g h g^{-1} \) is also in \( H \) whenever \( h \) is in \( H \).
The Duflo isomorphism is a concept in the field of mathematics, specifically in the study of Lie algebras and representation theory. Named after the mathematician Michel Duflo, this isomorphism establishes a deep connection between the functions on a Lie group and the representation theory of its corresponding Lie algebra.
Fitting's theorem, named after the mathematician W. Fitting, is a result in the field of group theory, specifically concerning the structure of finite groups. It provides important information about the composition of a finite group in terms of its normal subgroups and nilpotent components.
The Fontaine–Mazur conjecture is a significant conjecture in number theory, particularly in the areas of Galois representations and modular forms. Proposed by Pierre Fontaine and Bertrand Mazur in the 1990s, the conjecture relates to the solutions of certain Diophantine equations and the nature of Galois representations.
A **free ideal ring** is a concept from abstract algebra that relates to ring theory. Specifically, it refers to a certain kind of algebraic structure derived from a free set of generators. Let me explain it in more detail. ### Definitions: 1. **Ring**: A ring is a set equipped with two operations: addition and multiplication, satisfying certain axioms (such as associativity, distributivity, etc.).
The Freudenthal algebra, also known as the Freudenthal triple system, is a mathematical structure introduced by Hans Freudenthal in the context of nonlinear algebra. It is primarily used in the study of certain Lie algebras and has connections to exceptional Lie groups and projective geometry. A Freudenthal triple system is defined as a vector space \( V \) equipped with a bilinear product, which satisfies specific axioms.
A Gelfand ring is a specific type of ring that arises in the study of functional analysis and commutative algebra, particularly in the context of commutative Banach algebras. It is named after the mathematician I.M. Gelfand. A Gelfand ring is defined as follows: 1. **Commutative Ring**: A Gelfand ring is a commutative ring \( R \) that is also equipped with a topology.
Goldie's theorem, in the context of algebra and particularly concerning semigroups and group theory, pertains to the structure of certain algebraic objects. It is often discussed in relation to goldie dimensions and the growth of modules over rings.
The Goncharov conjecture is a hypothesis in the field of algebraic geometry and number theory, proposed by Russian mathematician Alexander Goncharov. It concerns the behavior of certain algebraic cycles in the context of motives, which are a central concept in modern algebraic geometry. Specifically, the conjecture deals with the relationships between Chow groups, which are groups that classify algebraic cycles on a variety, and their connection to motives.
The Gorenstein–Walter theorem is a result in the area of algebra, particularly in the study of Gorenstein rings and commutative algebra. It essentially characterizes certain types of Gorenstein rings. The theorem states that a finitely generated algebra over a field which has a Gorenstein ring structure is Cohen-Macaulay and that such rings have certain properties related to their module categories.
In ring theory, a branch of abstract algebra, the concept of "grade" often pertains to the structure of graded rings, which are rings that can be decomposed into a direct sum of abelian groups or modules indexed by integers or another grading set.
Grothendieck's connectedness theorem is a result in algebraic geometry that relates to the structure of schemes, particularly concerning the notion of connectedness in the context of the Zariski topology.
The Grothendieck Existence Theorem is a fundamental result in algebraic geometry that pertains to the construction of schemes and their coherent sheaves, particularly in the context of the development of the theory of stacks and the Grothendieck topology. In more detail, the theorem addresses the existence of certain kinds of algebraic objects, providing conditions under which a given formal object can be realized by a certain kind of "concrete" object.
The Group Isomorphism Problem is a computational problem in the field of algebra and computer science. It concerns the determination of whether two finite groups are isomorphic, meaning that there exists a bijection (one-to-one and onto mapping) between their elements that preserves the group operation.
A **hereditary ring** is a type of ring in the field of abstract algebra, particularly in ring theory. A ring \( R \) is called hereditary if every finitely generated module over \( R \) is a projective module. This is equivalent to saying that all submodules of finitely generated projective modules are also projective. In simpler terms, projective modules are those that resemble free modules in terms of their structure and properties.
The Hochschild–Mostow group is a concept from algebraic topology, particularly in the area of algebraic K-theory and homotopy theory. It is associated with the study of higher-dimensional algebraic structures and their symmetries.
A **hyperfinite field** typically refers to a concept in the realm of mathematical logic and model theory, particularly in the study of non-standard analysis and structures. It is often related to the idea of constructing fields that have properties akin to finite fields but with an infinite nature.
Idempotent analysis is a branch of mathematics and theoretical computer science that extends the concepts of traditional analysis using the framework of idempotent semirings. In idempotent mathematical structures, the operation of addition is replaced by a max operation (or another specific operation depending on the context), and the operation of multiplication remains similar to standard multiplication.
The term "inner form" can have different meanings depending on the context in which it is used. Here are a few interpretations based on various fields: 1. **Linguistics**: In linguistics, "inner form" can refer to the underlying meaning or semantic structure of a word or expression, as opposed to its "outer form," which is the phonetic or written representation. This concept is often discussed in relation to the relationship between language, thought, and reality.
In the context of ring theory, an **irreducible ideal** is a specific type of ideal in a ring that has certain properties.
The Isomorphism Extension Theorem is a result in the field of abstract algebra, particularly in the study of groups, rings, and modules. It provides a framework for extending certain structures while preserving their key properties. The theorem is often discussed in the context of group and module theory, where it deals with homomorphisms and their extensions.
Itô's theorem is a fundamental result in stochastic calculus, particularly in the context of stochastic processes involving Brownian motion. Named after Japanese mathematician Kiyoshi Itô, the theorem provides a method for finding the differential of a function of a stochastic process, typically a Itô process.
The term "Jacobi group" can refer to a specific mathematical structure in the field of algebra, particularly within the context of Lie groups and their representations. However, the name might be more commonly associated with Jacobi groups in the context of harmonic analysis on homogeneous spaces or in certain applications in number theory and geometry. In one interpretation, **Jacobi groups** are related to **Jacobi forms**.
A Jaffard ring is a concept in the field of functional analysis and operator theory, named after the mathematician Claude Jaffard. It is related to the study of certain types of algebras of operators, particularly those exhibiting specific algebraic and topological properties.
The Johnson-Wilson theory is a theoretical framework used in solid-state physics and condensed matter physics to describe the electronic structure of materials, particularly correlated electron systems like high-temperature superconductors and heavy fermion compounds. This theory builds on concepts from quantum mechanics and many-body physics. The key aspects of Johnson-Wilson theory include: 1. **Effective Hamiltonian**: The theory often employs model Hamiltonians that capture the essential interactions and correlations between electrons in a material.
The Karoubi conjecture is a hypothesis in the field of algebraic topology, particularly concerning the relationships between certain types of groups associated with topological spaces. Specifically, it relates to the K-theory of a space and the structure of its stable homotopy category. In more technical terms, the conjecture posits that every homotopy equivalence between simply-connected spaces induces an isomorphism on their stable homotopy categories.
The Kawamata–Viehweg vanishing theorem is a result in algebraic geometry that deals with the cohomology of certain coherent sheaves on projective varieties, particularly in the context of higher-dimensional algebraic geometry. It addresses conditions under which certain cohomology groups vanish, which is crucial for understanding the geometry of algebraic varieties and the behavior of their line bundles.
Kummer varieties are algebraic varieties associated with abelian varieties, specifically focusing on the quotient of a complex torus that arises from abelian varieties. More precisely, a Kummer variety is constructed from an abelian variety by identifying points that are negatives of each other.
The Kurosh problem, named after the Iranian mathematician Alexander Kurosh, is a well-known problem in group theory, particularly in the context of the structure of groups and their subgroups. The Kurosh problem concerns the characterization of a certain type of subgroup, namely, free products of groups.
Luna's slice theorem is a result in the field of algebraic geometry and it pertains to the study of group actions on algebraic varieties. Specifically, it deals with the situation where a group acts on a variety, and it provides a way to understand the local structure of the variety at points with a particular kind of symmetry.
A Marot ring is a type of mathematical structure used in the study of algebraic topology, specifically in the context of homotopy theory and the theory of operads. It is named after the mathematician Marot, who contributed to the development of these concepts. In more detail, a Marot ring can be seen as a certain kind of algebraic object that exhibits properties related to the arrangement and composition of topological spaces or other algebraic structures.
A **metabelian group** is a specific type of group in the field of group theory. A group \( G \) is called metabelian if its derived subgroup (also known as the commutator subgroup) is abelian.
A metacyclic group is a specific type of group in group theory, which is a branch of mathematics. More precisely, it is a particular kind of solvable group that has a structure related to cyclic groups. A group \( G \) is called metacyclic if it has a normal subgroup \( N \) that is cyclic, and the quotient group \( G/N \) is also cyclic.
The Milnor–Moore theorem is a key result in the field of differential topology and algebraic topology, specifically concerning the structure of certain classes of smooth manifolds. Named after mathematicians John Milnor and John Moore, the theorem provides a characterization of the relationship between the algebra of smooth functions on a manifold and the algebra of its vector fields.
Naimark equivalence is a concept in functional analysis and operator theory that relates to the representation of certain kinds of operator algebras, specifically commutative C*-algebras. The concept is named after the mathematician M.A. Naimark.
Nakayama's conjecture is a significant hypothesis in the field of algebra, specifically within commutative algebra and the study of Noetherian rings. Formulated by Takashi Nakayama in the 1950s, it deals with the behavior of certain types of modules over local rings.
The Neukirch–Uchida theorem is a result in algebraic number theory, specifically concerning the relationship between certain Galois groups and the structure of algebraic field extensions.
In mathematics, specifically in abstract algebra, an **opposite ring** is a concept that arises when considering the structure of rings in a different way. If \( R \) is a ring, the **opposite ring** \( R^{op} \) (also sometimes denoted as \( R^{op} \) or \( R^{op} \)) is defined with the same underlying set as \( R \), but with the multiplication operation reversed.
An **ordered semigroup** is a mathematical structure that combines the concepts of semigroups and ordered sets.
Orthomorphism is a term primarily used in the context of mathematics and particularly in the study of algebraic structures. It can refer to a type of homomorphism—a structure-preserving map between two algebraic structures—specifically when dealing with groups or other algebraic systems. In a more general sense, an orthomorphism can denote a specific kind of morphism that preserves certain properties or structures in a more 'orthogonal' way.
In the context of group theory, a **permutation representation** is a way of representing a group as a group of permutations. Specifically, if \( G \) is a group, a permutation representation of \( G \) is a homomorphism from \( G \) to the symmetric group \( S_n \), which is the group of all permutations of a set of \( n \) elements.
Petersson algebra, named after the mathematician Harold Petersson, is a specific algebraic structure that arises in the context of modular forms and number theory. It is particularly relevant in the study of modular forms of several variables and their associated spaces. In the context of modular forms, Petersson algebra describes the action of certain differential operators that provide a natural way to analyze and construct modular forms.
In algebraic topology, a Postnikov square is a geometric construction that provides an important method for studying topological spaces up to homotopy. Specifically, it is used to break down a space into simpler pieces that are easier to analyze in terms of their homotopy types.
"Preradical" might refer to a concept or term that is not widely recognized in mainstream discourse as of my last training cut-off in October 2023. It could potentially be a term used in specific academic fields, niche discussions, or could be a typographical error or shorthand for something else, such as "pre-radical" in a political or ideological context.
In various contexts, the term "primary extension" can have different meanings. Here are a few interpretations based on different fields: 1. **Mathematics**: In algebra, particularly in the study of fields and rings, a "primary extension" might refer to an extension of fields that preserves certain properties of the original field. The concept of field extensions is fundamental in algebra, and primary extensions might involve specific types of extensions such as algebraic or transcendental extensions.
In the context of finite fields (also known as Galois fields), a **primitive element** is an element that generates the multiplicative group of the field. To understand this concept clearly, let's start with some basics about finite fields: 1. **Finite Fields**: A finite field \( \mathbb{F}_{q} \) is a field with a finite number of elements, where \( q \) is a power of a prime number, i.e.
The term "principal factor" can refer to various concepts depending on the context, such as mathematics, finance, or other fields. Here are a few interpretations in different contexts: 1. **Mathematics**: In the context of number theory, a principal factor may refer to the largest prime factor of a given integer.
Protorus is a term that could refer to different concepts depending on the context, but it is not widely recognized or standardized in a specific field as of my last knowledge update in October 2023. It might be related to mathematical, physical, or engineering concepts involving toroidal shapes or structures. In some contexts, it might also refer to software, a company name, or a specific project.
Quantum affine algebras are a class of mathematical objects that arise in the area of quantum algebra, which blends concepts from quantum mechanics and algebraic structures. To understand quantum affine algebras, it's helpful to break down the components involved: 1. **Affine Algebras**: These are a type of algebraic structure that generalize finite-dimensional Lie algebras. An affine algebra can be thought of as an infinite-dimensional extension of a Lie algebra, which incorporates the concept of loops.
In algebraic geometry and related fields, a **quasi-compact morphism** is a type of morphism of schemes or topological spaces that relates to the compactness of the images of certain sets. A morphism of schemes \( f: X \to Y \) is called **quasi-compact** if the preimage of every quasi-compact subset of \( Y \) under \( f \) is quasi-compact in \( X \).
A quasi-triangular quasi-Hopf algebra is a generalization of the concept of a quasi-triangular Hopf algebra. These structures arise in the field of quantum groups and related areas in mathematical physics and representation theory.
Regev's theorem is a result from the field of lattice-based cryptography, specifically concerning the hardness of certain mathematical problems in lattice theory. The theorem, established by Oded Regev in 2005, demonstrates that certain problems in lattices, such as the Learning with Errors (LWE) problem, are computationally hard, meaning they cannot be efficiently solved by any known classical algorithms.
In the context of mathematics, specifically in the fields of algebra and topology, a "regular extension" can refer to different concepts depending on the area of study. Here are a couple of interpretations of the term: 1. **Field Theory**: In field theory, a regular extension can refer to an extension of fields that behaves well under certain algebraic operations.
In mathematics, particularly in the field of algebra and number theory, the term "residual property" can refer to several concepts depending on the context. However, it is not a standard term and may not have a single, universally accepted definition across branches of mathematics.
Rigid cohomology is a relatively new and sophisticated theory in the field of arithmetic geometry, developed primarily by Bhargav Bhatt and Peter Scholze. It serves as a tool to study the properties of schemes over p-adic fields, with a focus on their rigid analytic aspects. Rigid cohomology generalizes several classical notions in algebraic geometry and offers a framework for understanding phenomena in the realm of p-adic Hodge theory.
In algebraic number theory, a **ring class field** is an important concept related to algebraic number fields and their class groups. To understand ring class fields, we first need to introduce a few key concepts: 1. **Algebraic Number Field:** An algebraic number field is a finite field extension of the rational numbers \(\mathbb{Q}\). It can be represented as \(\mathbb{Q}(\alpha)\) for some algebraic integer \(\alpha\).
A **ring spectrum** is a concept from stable homotopy theory, which is a branch of algebraic topology. It generalizes the idea of a ring in the context of stable homotopy categories, allowing us to study constructions involving stable homotopy groups and cohomology theories in a coherent way. In more technical terms, a ring spectrum is a spectrum \( R \) that comes equipped with multiplication and unit maps that satisfy certain properties.
The SBI Ring is a digital payment solution developed by the State Bank of India (SBI) that allows users to make payments using a physical ring. The ring is equipped with NFC (Near Field Communication) technology, enabling users to make contactless payments at point-of-sale terminals by simply tapping their ring.
The Schreier conjecture is a conjecture in the field of group theory, specifically concerning the properties of groups of automorphisms. It was proposed by Otto Schreier in 1920. The conjecture states that for every infinite group \( G \) of automorphisms, the rank of the group of automorphisms \( \text{Aut}(G) \) is infinite.
A **Schreier domain** is a specific type of integral domain in the field of algebra, particularly in the study of ring theory. By definition, a domain is a commutative ring with unity in which there are no zero divisors. A Schreier domain is characterized by certain structural properties that relate to its ideals and factorizations.
The Schreier refinement theorem is a result in group theory that deals with the relationship between subgroups and normal series of a group. It provides criteria for refining a normal series of groups, allowing for more structured decompositions of groups into simpler components. The theorem is primarily used in the study of group extensions and solvable groups.
A semiprimitive ring is a type of ring in algebra that has specific properties related to its ideal structure. More formally, a ring \( R \) is called semiprimitive if it is a direct sum of simple Artinian rings, or equivalently, if its Jacobson radical is zero, i.e., \[ \text{Jac}(R) = 0.
Shafarevich's theorem, often discussed in the context of algebraic number theory, specifically addresses the solvability of Galois groups of field extensions. The theorem essentially states that under certain conditions, a Galois extension of a number field can have a Galois group that is solvable.
The term "Slender group" generally refers to a specific type of mathematical group in the context of group theory, particularly in the area of algebra. More formally, a group \( G \) is called a slender group if it satisfies certain conditions regarding its subgroups and representations. In particular, slender groups are often defined in the context of topological groups or the theory of abelian groups.
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Not to be confused with algebra over a field, which is a particular algebraic structure studied within algebra.