Ring theory

Ring theory is a branch of abstract algebra that studies algebraic structures known as rings. A ring is a set equipped with two operations that generalize the arithmetic operations of addition and multiplication. Specifically, a ring \( R \) is defined by the following properties: 1. **Addition**: - The set \( R \) is closed under addition.

A finite ring is a ring that contains a finite number of elements. In abstract algebra, a ring is defined as a set equipped with two binary operations: addition and multiplication, which satisfy certain properties. Specifically, a ring must satisfy the following axioms: 1. **Additive Identity**: There exists an element \(0\) such that \(a + 0 = a\) for all elements \(a\) in the ring.

A Galois ring is a type of algebraic structure related to the field of Galois theory and finite fields. It generalizes the concept of a finite field and is particularly useful in coding theory and other areas of mathematics.

Geometric algebra is a mathematical framework that extends traditional algebra by incorporating geometric concepts. It is a unifying language for describing geometric transformations and relationships, merging algebraic and geometric perspectives. Here are some key aspects of geometric algebra: 1. **Multivectors**: In geometric algebra, quantities such as points, lines, planes, and volumes are represented as multivectors.

In geometry, particularly in the context of polyhedral modeling and computer graphics, a "blade" typically refers to a flat, planar surface that forms part of a three-dimensional object. However, the term can have specific meanings in different fields or contexts. For example, in the context of 3D modeling, blades may refer to the flat surfaces that make up the faces of a polyhedron or a more complex geometric shape.

Vector algebra and geometric algebra are two mathematical frameworks used to study and manipulate vectors and their properties, but they have different focuses and methodologies. Below is a comparison of the two: ### Definition: - **Vector Algebra**: This is a branch of algebra that deals with vectors, which are objects that have both magnitude and direction. It typically involves operations such as addition, scalar multiplication, dot product, and cross product.

Conformal geometric algebra (CGA) is a mathematical framework that extends traditional geometric algebra to include conformal transformations, which preserve angles but not necessarily distances. CGA is particularly useful in computer graphics, robotics, and advanced physics, as it provides a unified algebraic language for handling geometric concepts. ### Key Features of Conformal Geometric Algebra 1.

Formal moduli refers to a branch of algebraic geometry that studies families of algebraic objects (such as varieties or schemes) over a base, typically in a formal or non-archimedean setting. This concept is often used in the context of deformation theory and moduli problems, where one is interested in understanding how objects of a given type can be continuously deformed into one another.

Gauge theory gravity is a theoretical framework that seeks to describe gravity in terms of gauge theories, which are a class of field theories where symmetries play a crucial role. In conventional general relativity, gravity is described as a geometric property of spacetime, expressed through the curvature of the spacetime manifold. In contrast, gauge theories are typically formulated using fields that are invariant under certain transformations (gauge transformations).

Outermorphism is a concept in the field of mathematics, specifically in category theory, which deals with the structure and relationships between different mathematical objects. While "outermorphism" is not a standard term widely recognized in mathematics, it may refer to a specific type of morphism that relates to certain structures or transformations in a broader context. In general, the term "morphism" in category theory refers to a structural-preserving map between two objects.

Plane-based Geometric Algebra is a specialized framework within the broader field of Geometric Algebra (GA) that focuses on vector spaces defined by planes. Geometric Algebra itself is an algebraic system that extends linear algebra and provides a unified way to handle geometric transformations, including rotations and reflections, as well as more complex geometrical relations. In Plane-based Geometric Algebra, the primary elements are typically oriented around two-dimensional planes, allowing for relevant operations defined in that context.

The term "plane of rotation" refers to the imaginary plane in which the rotation of an object occurs. It is a geometric concept used in various fields, including physics, engineering, and mathematics, to describe the orientation and axis about which an object rotates. ### Key Points: 1. **Rotational Motion**: In the context of rotational motion, the plane of rotation is typically perpendicular to the axis of rotation.

Quadric geometric algebra refers to an extension of geometric algebra that is specifically designed to handle geometric and algebraic structures related to quadrics, which are second-degree algebraic surfaces. Quadrics can be represented in various forms, such as ellipsoids, hyperboloids, paraboloids, and other related shapes, and they play a significant role in both geometry and physics.

The Riemann–Silberstein vector is a mathematical construct used in the context of electromagnetic theory. It provides a unified way to represent electric and magnetic fields. Named after Bernhard Riemann and Hans Silberstein, the vector is particularly useful in theoretical physics, especially in the study of electromagnetic waves and their propagation.

In mathematics, specifically in vector calculus, the term "rotor" often refers to the **curl** of a vector field. The curl measures the tendency of a vector field to induce rotation at a point in space.

Universal Geometric Algebra (UGA) is a mathematical framework that generalizes various geometric concepts and structures using the tools of algebra. It combines elements of linear algebra, multilinear algebra, and geometric reasoning to provide a unified language and method for analyzing geometric problems. At its core, UGA extends traditional ideas of geometric algebra to develop a more comprehensive system that can describe various geometric entities, such as points, lines, planes, and higher-dimensional analogs.

In ring theory, which is a branch of abstract algebra, an **ideal** is a specific subset of a ring that has particular properties allowing it to be used in the construction of quotient rings and in the study of ring homomorphisms. ### Definition: Let \( R \) be a ring (with unity, but this requirement can be relaxed in some contexts).

In abstract algebra, particularly in the context of ring theory, a **prime ideal** is a special type of ideal that has important properties related to the structure of rings.

The Ascending Chain Condition (ACC) on principal ideals is a property related to the structure of a ring in the context of ideal theory. Specifically, a ring \( R \) satisfies the ACC on principal ideals if any ascending chain of principal ideals eventually stabilizes.

In the context of algebra, particularly in ring theory and module theory, an **augmentation ideal** is a specific ideal associated with a group ring or a similar algebraic structure. ### Definition 1. **Group Ring Context**: If \( k \) is a field and \( G \) is a group, the group ring \( k[G] \) consists of formal sums of elements of \( G \) with coefficients in \( k \).

In the context of algebraic number theory, a **fractional ideal** is a generalization of the notion of an ideal in a ring. Specifically, fractional ideals are particularly useful in the study of Dedekind domains and more generally in the structure of arithmetic in number fields. ### Definitions and Properties 1. **Integral Domain**: First, consider a domain \( R \), typically a Dedekind domain, which is an integral domain where every nonzero proper prime ideal is maximal.

In order theory, an **ideal** is a specific subset of a partially ordered set (poset) that captures a certain type of "lower" structure.

The ideal class group is an important concept in algebraic number theory, particularly in the study of ring theory and algebraic integers. It provides a way to measure the failure of unique factorization in the ring of integers of a number field.

In algebra, particularly in the context of commutative rings, the term "ideal quotient" refers to a concept that is used to define the relationship between ideals.

The Jacobian ideal is a concept in algebraic geometry and commutative algebra, associated with a polynomial ring and its derivatives. It is particularly important in the study of algebraic varieties and singularities.

Krull's theorem is a result in commutative algebra that pertains to the structure of integral domains, specifically regarding the heights of prime ideals in a Noetherian ring. The theorem states: In a Noetherian ring (or integral domain), the height of a prime ideal \( P \) is less than or equal to the number of elements in any generating set of the ideal \( P \).

In the context of ring theory, a nil ideal is an ideal \( I \) of a ring \( R \) such that every element \( x \) in \( I \) is nilpotent.

In abstract algebra, specifically in the study of rings, a **nilpotent ideal** is an ideal such that there exists some positive integer \( n \) for which the \( n \)-th power of the ideal is equal to the zero ideal.

In algebra, particularly in commutative algebra, the radical of an ideal is a fundamental concept used to study the properties of ideals and rings.

The term "real radical" can refer to a few different concepts depending on the context, but in general mathematics and algebra, a "radical" typically refers to an operation that involves roots, such as square roots, cube roots, etc. When we say "real radical," we are usually indicating that we are dealing specifically with real numbers rather than complex numbers. For example: - A real radical of the form \(\sqrt{x}\) is defined for non-negative real numbers \(x\).

In mathematics, an algebra extension (often referred to as a "field extension" in a specific context) typically involves expanding a given algebraic structure to include additional elements that satisfy certain properties or relationships. 1. **Field Extensions**: In the context of field theory, a field extension is a larger field that contains a smaller field as a subfield.

Artin algebras are a class of associative algebras that have several important properties in representation theory and algebra. Specifically, an Artin algebra is defined as a finite-dimensional algebra over a field that satisfies certain conditions. Here are some key features of Artin algebras: 1. **Finite Length**: An Artin algebra has the property that as a module over itself, it has finite length. This means that it has a composition series with a finite number of simple submodules.

An Artinian ring is a type of ring in ring theory, which is a branch of abstract algebra. A ring \( R \) is called Artinian if it satisfies the descending chain condition (DCC) on ideals. This means that any descending chain of ideals in \( R \): \[ I_1 \supseteq I_2 \supseteq I_3 \supseteq \ldots \] eventually stabilizes, i.e.

An associated graded ring is a construction in algebra that arises in the study of filtered rings, which are rings equipped with a specified filtration.

An **Azumaya algebra** is a specific type of algebra over a commutative ring that behaves like a matrix algebra over a field in a certain sense, while being more general. Formally, an Azumaya algebra is a sheaf of algebras that satisfies a particular condition related to the notion of being "frobenius". More precisely, let \( R \) be a commutative ring.

A Baer ring is a specific type of algebraic structure in the field of ring theory, which is a branch of abstract algebra.

A binomial ring is a specific type of ring in mathematics that is characterized by its elements having a form that resembles binomials.

Biquaternions extend the concept of quaternions to include a second imaginary unit, often denoted as \( j \), in addition to the standard quaternion imaginary unit \( i \).

Central simple algebras are a fundamental concept in algebra, particularly in the study of algebraic structures over fields. Let's break down what central simple algebras are: 1. **Algebra**: In the context of central simple algebras, an algebra refers to a vector space equipped with a multiplication operation that is associative and distributes over vector addition.

Clifford algebras are a type of associative algebra that arise naturally in various areas of mathematics and physics, particularly in the study of geometric transformations and spinors. The classification of Clifford algebras is typically done based on the dimension of the underlying vector space and the signature of the quadratic form used to define them.

Clifford algebra is a mathematical structure that extends the concept of vector spaces and inner products into a more generalized algebraic framework. It is named after the mathematician William Kingdon Clifford, who developed the concept in the 19th century. ### Definition A Clifford algebra is generated from a vector space \( V \) equipped with a quadratic form.

In mathematics, particularly in the context of abstract algebra, a **coherent ring** is a type of ring that satisfies a specific property related to its finitely generated ideals. Specifically, a ring \( R \) is coherent if every finitely generated ideal of \( R \) is finitely presented.

The Dedekind–Hasse norm is a concept from algebraic number theory that concerns the behavior of norms of ideals in the context of Dedekind domains. A Dedekind domain is a specific type of integral domain that satisfies certain properties, including being Noetherian, integrally closed, and having the property that every nonzero prime ideal is maximal.

In the context of noncommutative ring theory, the "depth" of a ring can be understood analogously to the depth of a module or a commutative ring. However, since you're asking about noncommutative subrings, it's important to clarify a few concepts. 1. **Depth in Commutative Rings**: In commutative algebra, the depth of a ring is defined in terms of the length of the longest regular sequence of its ideals.

In ring theory, a branch of abstract algebra, divisibility refers to a relation between elements of a ring that generalizes the familiar notion of divisibility from the integers. In more formal terms, let \( R \) be a ring and let \( a, b \in R \).

Division algebra is a type of algebraic structure where division is possible, except by zero. More formally, a division algebra is a vector space over a field \( F \) equipped with a bilinear multiplication operation that satisfies the following conditions: 1. **Non-Associativity or Associativity**: In a general division algebra, multiplication can be either associative or non-associative. If it is associative, the algebra is called an associative division algebra.

A **division ring** is a type of algebraic structure in abstract algebra. It is similar to a field, but with a key difference regarding the requirement for multiplication. Here are the main characteristics of a division ring: 1. **Set with Two Operations**: A division ring consists of a set \( D \) equipped with two binary operations: addition (+) and multiplication (·).

In mathematics, particularly in the area of ring theory, the concept of a fixed-point subring can arise in various contexts. While the term "fixed-point subring" may not have a universally standardized definition, it can be understood in the framework of fixed points in algebraic structures. A fixed-point of a function is an element that is mapped to itself by that function.

A glossary of ring theory includes key terms and concepts that are fundamental to the study of rings in abstract algebra. Here are some important terms and their definitions: 1. **Ring**: A set \( R \) equipped with two binary operations, typically called addition and multiplication, satisfying certain properties (e.g., closure, associativity, distributivity, existence of an additive identity, and existence of additive inverses).

The term "Goldman domain" is not widely recognized in common academic or scientific literature as of my last knowledge update in October 2023. It is possible that it refers to a concept related to finance or economics, particularly if it is associated with Goldman Sachs, a well-known investment banking firm, but without further context, it’s hard to provide a precise definition.

A graded ring is a type of ring that is decomposed into a direct sum of abelian groups (or modules) based on their degree, with specific rules about how the elements from different degrees interact with one another under multiplication.

In ring theory, an element \( a \) of a ring \( R \) is said to be **idempotent** if it satisfies the condition: \[ a^2 = a. \] In other words, when you multiply the element by itself, you get the same element back. Idempotent elements play a significant role in various areas of algebra, particularly in the study of ring structure and module theory.

Jacobson's conjecture is a conjecture in the field of algebra, specifically relating to rings and their structure. It proposes that for a finitely generated ring \( R \) over a field, the Jacobson radical \( J(R) \) has certain characteristics.

The Jacobson radical is a concept that arises in the context of ring theory, a branch of abstract algebra. It is a particular ideal associated with a ring, which captures information about the ring's structure in relation to simple modules and semisimplicity. Here are the key points regarding the Jacobson radical: 1. **Definition**: The Jacobson radical \( J(R) \) of a ring \( R \) is defined as the intersection of all maximal left ideals of \( R \).

Kaplansky's conjectures refer to a set of conjectures proposed by the mathematician David Kaplansky, primarily in the area of ring theory and algebra. One of the most famous is related to the structure of rings, particularly concerning division rings and their fields of fractions. 1. **Kaplansky's Division Ring Conjecture**: This conjecture posits that every division ring that is finitely generated as a module over its center is a field.

A **Kleinian integer** is a concept from the field of hyperbolic geometry and complex analysis. Specifically, it is defined as a number that can be expressed in the form \(a + b\sqrt{-d}\), where \(a\) and \(b\) are integers and \(d\) is a positive integer that is not a perfect square.

The Köthe conjecture is a mathematical conjecture related to the field of functional analysis, particularly in the context of Banach spaces. Proposed by the German mathematician Heinrich Köthe in the mid-20th century, the conjecture concerns the structure of certain types of Banach spaces known as Köthe spaces, which are defined in terms of sequence spaces and their properties.

Leavitt path algebras are a class of algebras that arise from directed graphs (or quivers) and are named after the mathematician William G. Leavitt, who studied related structures in the context of rings. **Definition:** A Leavitt path algebra is constructed from a directed graph \( E \) and involves both paths in the graph and the concept of vertices and edges.

A Loewy ring is a type of algebraic structure that arises in the study of representation theory and module theory. Specifically, it is a class of rings that have certain desirable properties regarding their modules. Loewy rings are defined in the context of "Loewy series," which are derived series of a module that break it down into a sequence of submodules.

In abstract algebra, a **maximal ideal** is a specific type of ideal within a ring. To define it, let's first recall some basic concepts related to rings and ideals: 1. **Ring**: A set equipped with two binary operations, typically called addition and multiplication, satisfying certain properties (like associativity, distributivity, etc.). 2. **Ideal**: A subset of a ring that absorbs multiplication by elements from the ring and is closed under addition.

A **monoid ring** is an algebraic structure that combines concepts from both ring theory and the theory of monoids. Specifically, it is formed from a monoid \( M \) and a ring \( R \). Here's a more detailed breakdown of what this means: 1. **Monoid**: A monoid is a set \( M \) equipped with a single associative binary operation (let's denote it by \( \cdot \)) and an identity element \( e \).

Nakayama algebra is a type of algebra that arises in the context of representation theory and, more specifically, in the study of finite-dimensional algebras over a field. Nakayama algebras are named after the mathematician Tadao Nakayama and are characterized by their structural properties which relate to the representation theory of algebras.

A necklace ring, also known as a "necklace pendant ring" or "ring necklace," is a type of jewelry that combines elements of both rings and necklaces. Typically, a necklace ring consists of a ring or band that is worn as a pendant on a chain or cord. The design can vary widely, featuring gemstones, intricate metalwork, or unique shapes. People often wear necklace rings for various reasons, including fashion statements, sentimental value, or as part of cultural or religious traditions.

In mathematics, particularly in the field of algebra, a **nilpotent algebra** generally refers to an algebraic structure where the elements exhibit certain properties related to nilpotency. While the term can refer to different types of structures depending on the context, the most common interpretation relates to **nilpotent operators** or **nilpotent matrices** in linear algebra.

A Noetherian ring is a specific type of ring in algebra that satisfies a property related to the concept of ideal containment. A ring \( R \) is called Noetherian if it satisfies any of the following equivalent conditions: 1. **Ascending Chain Condition on Ideals (ACC)**: Every ascending chain of ideals in \( R \) stabilizes.

Non-integer bases of numeration refer to number systems that use bases that are not whole numbers or integers. Most commonly, we are familiar with integer bases like base 10 (decimal), base 2 (binary), and base 16 (hexadecimal). However, bases can also be fractional or irrational. ### Key Concepts: 1. **Base Representation**: In a base \( b \) system, numbers are represented using coefficients for powers of \( b \).

A **noncommutative ring** is a type of algebraic structure that generalizes some properties of familiar number systems, like the integers or polynomials, but allows for multiplication where the order of the factors matters. In other words, in a noncommutative ring, it is possible for the product of two elements \( a \) and \( b \) to differ from the product \( b \) and \( a \); that is, \( ab \neq ba \).

A **noncommutative unique factorization domain (UFD)** is a generalization of the concept of a unique factorization domain in commutative algebra, extended to the realm of noncommutative algebra. In the context of commutative algebra, a unique factorization domain is an integral domain in which every non-zero non-unit element can be factored uniquely (up to order and units) into irreducible elements.

In ring theory, an *order* is a specific type of subset of a ring that behaves like the integers within that ring structure. More formally, if \( R \) is a ring and \( S \) is a subset of \( R \), we say that \( S \) is an order in \( R \) if: 1. \( S \) is a subring of \( R \) (i.e.

The Ore condition, named after the Norwegian mathematician, mathematician O. Ore, refers to a set of criteria in algebra that help identify whether a certain type of ring is integrally closed. More specifically, it is used to determine whether a finitely generated commutative algebra over a field is integrally closed in its field of fractions.

The Ore extension, named after the mathematician Ole Johan Dahl Ore, is a concept in algebra that pertains to the extension of rings and modules. In particular, it is used to construct new rings from a given ring by adding new elements and defining new operations. The most common application of Ore extensions occurs in the context of noncommutative algebra, where it is used to form the Ore localization of a polynomial ring. This involves extending a ring by introducing new elements that satisfy specific relations concerning multiplication.

A *partially ordered ring* is a mathematical structure that combines the properties of a ring and a partially ordered set. To elaborate, a structure \( (R, +, \cdot) \) is called a partially ordered ring if it satisfies the following conditions: 1. **Ring Structure**: - \( (R, +) \) is an abelian group, which means that addition is commutative, associative, and each element has an additive inverse.

In the context of ring theory, a branch of abstract algebra, a **perfect ring** is a specific type of ring that has certain characteristics relating to its structure, particularly concerning ideals and their relations to other elements in the ring.

A Poisson ring is an algebraic structure that combines aspects of both ring theory and Poisson algebra. Specifically, a Poisson ring is a commutative ring \( R \) equipped with a bilinear operation called the Poisson bracket, denoted \(\{ \cdot, \cdot \}\), that satisfies certain properties.

A **polynomial identity ring**, often denoted as \( R[x] \), is a specific type of ring formed by polynomials with coefficients from a ring \( R \). Here's a breakdown of the concepts involved: 1. **Polynomial Ring**: Given a ring \( R \), the polynomial ring \( R[x] \) is the set of all polynomials in the variable \( x \) with coefficients in \( R \).

In the context of mathematics, particularly in number theory and abstract algebra, a **prime element** (or simply a prime) is an element of an integral domain (a type of ring) that satisfies certain properties.

In ring theory, a branch of abstract algebra, a **prime ring** is a specific type of ring with certain properties that resemble those of prime numbers in number theory. A ring \( R \) is called **prime** if it is not the zero ring (i.e.

In mathematics, particularly in abstract algebra, the product of rings refers to a construction that combines two or more rings to form a new ring. There are different ways to define the product of rings, but the most common definition is that of the direct product (or Cartesian product) of rings.

The projective line over a ring \( R \), denoted as \( \mathbb{P}^1(R) \), is an important construction in algebraic geometry and commutative algebra. It extends the concept of the projective line over a field to the context of a more general ring.

In mathematics, particularly in the field of functional analysis and the study of operator algebras, a *quasiregular element* typically refers to an element of a Banach algebra or a more general algebraic structure that behaves somewhat like an invertible element, but not quite.

In abstract algebra, a **quotient ring** (or factor ring) is a construction that allows you to create a new ring from a given ring by partitioning it into cosets of a subring. More formally, let \( R \) be a ring and \( I \) be a two-sided ideal of \( R \).

In the context of ring theory, the term "radical" can refer to various concepts depending on the specific type of ring or the structure being considered. Here are some common types of radicals associated with rings: 1. **Nilradical**: The nilradical of a ring \( R \) is the ideal consisting of all nilpotent elements of \( R \).

A rank ring is a concept that can refer to various notions in different fields, such as mathematics, computer science, and even in organizational contexts. However, one specific use of "rank ring" relates to abstract algebra, particularly in the context of representation theory and algebraic structures. In the context of algebra, a **rank ring** typically refers to a ring that classifies linear transformations of vector spaces with specific properties.

In ring theory, a branch of abstract algebra, a **reduced ring** is a type of ring in which there are no non-zero nilpotent elements. A nilpotent element \( a \) in a ring \( R \) is defined as an element such that for some positive integer \( n \), \( a^n = 0 \). In simpler terms, if \( a \) is nilpotent, then raising it to some power eventually results in zero.

In the context of algebra, particularly in commutative algebra, a **regular ideal** typically refers to an ideal that satisfies certain properties relevant to the dimension theory of rings and algebraic geometry.

A **regular local ring** is a specific type of local ring that has a well-behaved structure in relation to its maximal ideal and its associated residue field. To define it more precisely: 1. **Local Ring**: A local ring \( R \) is a commutative ring with a unique maximal ideal \( \mathfrak{m} \).

The ring of integers, commonly denoted as \(\mathbb{Z}\), is the set of all whole numbers that includes positive integers, negative integers, and zero.

The Rosati involution is an important concept in the context of the theory of abelian categories and algebraic geometry, particularly in the study of coherent sheaves and moduli problems. It is a way to define a certain kind of duality between objects in a category, especially in relation to vector bundles on algebraic varieties or coherent sheaves on schemes.

A semiprime ring is a specific type of ring in the field of abstract algebra.

Simple algebra, often referred to in the context of universal algebra, is a branch of mathematics that studies algebraic structures in a general way. Universal algebra focuses on understanding the common properties and relationships between different algebraic structures, such as groups, rings, fields, lattices, and so on, rather than just specific examples. ### Key Concepts in Universal Algebra: 1. **Algebraic Structures**: These are sets equipped with operations that satisfy certain properties.

In mathematics, specifically in the field of abstract algebra, a **simple ring** is a non-zero ring \( R \) that has no non-trivial two-sided ideals. More formally, a ring \( R \) is simple if: 1. \( R \neq \{ 0 \} \) (the zero ring). 2. The only two-sided ideals of \( R \) are \( \{ 0 \} \) and \( R \) itself.

In mathematics, a square-free element is an integer or a polynomial that is not divisible by the square of any prime number (in the case of integers) or not divisible by the square of any irreducible polynomial (in the case of polynomials). ### For Integers: An integer \( n \) is square-free if there is no prime \( p \) such that \( p^2 \) divides \( n \).

A **subring** is a concept in abstract algebra, particularly in the study of ring theory. A subring is a subset of a ring that is itself a ring under the same operations (addition and multiplication) defined in the larger ring. To formally define a subring, let’s consider a ring \( R \) with two binary operations: addition \( + \) and multiplication \( \cdot \).

In the context of module theory, a **torsion-free module** is a specific type of module over a ring that satisfies certain properties with respect to torsion elements.

A triangular matrix ring is a specific type of matrix ring made up of upper or lower triangular matrices over a given ring. More formally, let's define it in a bit more detail. ### Definition: 1. **Triangular Matrices**: - An **upper triangular matrix** is a square matrix where all entries below the main diagonal are zero.

The Weyl algebra, typically denoted \( A_n \), is a type of non-commutative algebra that plays a significant role in various areas of mathematics, particularly in algebraic geometry, representation theory, and mathematical physics. Specifically, the Weyl algebra is defined over a field (often the field of complex numbers or rational numbers) and is generated by polynomial rings in several variables subject to certain relations.