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In mathematics, ring theory is the study of rings, algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as homological properties and polynomial identities. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible In Mathematics, for any Lie algebra L one can construct its universal enveloping algebra U ( L) Homological algebra is the branch of Mathematics which studies homology in a general algebraic setting

Commutative rings are much better understood than noncommutative ones. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property Due to its intimate connections with algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, their theory, which is considered to be part of commutative algebra and field theory rather than of general ring theory, is quite different in flavour from the theory of their noncommutative counterparts. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In Mathematics, algebraic number theory is a major branch of Number theory which studies the Algebraic structures related to Algebraic integers Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings A fairly recent trend, started in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups, attempts to turn the situation around and build the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on (non-existent) 'noncommutative spaces'. Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the In Mathematics and Theoretical physics, quantum groups are certain Noncommutative algebras that first appeared in the theory of Quantum integrable systems The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

Please refer to the glossary of ring theory for the definitions of terms used throughout ring theory. Ring theory is the branch of Mathematics in which rings are studied that is structures supporting both an Addition and a Multiplication operation

Contents

History

The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. In Mathematics, especially in the field of Abstract algebra, a polynomial ring is a ring formed from the set of Polynomials in one or more variables This article deals with the ring of complex numbers integral over Z. Furthermore, the appearance of hypercomplex numbers in the mid-nineteenth century undercut the pre-eminence of fields in mathematical analysis. The term hypercomplex number has been used in Mathematics for the elements of algebras that extend or go beyond Complex number arithmetic In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

Richard Dedekind introduced the concept of a ring. Julius Wilhelm Richard Dedekind ( October 6, 1831 &ndash February 12, 1916) was a German mathematician who did important

The term ring (Zahlring) was coined by David Hilbert in the article Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. David Hilbert ( January 23, 1862 &ndash February 14, 1943) was a German Mathematician, recognized as one of the most 4, 1897.

The first axiomatic definition of a ring was given by Adolf Fraenkel in an essay in Journal für die reine und angewandte Mathematik (A. Abraham Halevi (Adolf Fraenkel (אברהם הלוי (אדולף פרנקל February 17 1891 Munich, Germany – October 15 Crelle's Journal, or just Crelle, is the common name for a leading German -language Mathematical journal, the Journal für L. Crelle), vol. 145, 1914.

In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper Ideal Theory in Rings. Amalie Emmy Noether, ˈnøːtɐ (23 March 1882 – 14 April 1935 was a German Mathematician known for her groundbreaking contributions to Abstract algebra and In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property

Elementary introduction

Definition

Formally, a ring is an Abelian group (R, +), together with a second binary operation * such that for all a, b and c in R,

a * (b * c) = (a * b) * c
a * (b + c) = (a * b) + (a * c)
(a + b) * c = (a * c) + (b * c)

also, if there exists a multiplicative identity in the ring, that is, an element e such that for all a in R,

a * e = e * a = a

then it is said to be a ring with unity. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two The number 1 is a common example of a unity.

It is simple to show that any ring in which e = 0 must have just one element; any such ring is called a zero ring.

Rings that sit inside other rings are called subrings. In Mathematics, a subring is a Subset of a ring, which contains the Multiplicative identity and is itself a ring under the same Binary operations Maps between rings which respect the ring operations are called ring homomorphisms. In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication Rings, together with ring homomorphisms, form a category (the category of rings). In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity and whose Morphisms Closely related is the notion of ideals, certain subsets of rings which arise as kernels of homomorphisms and can serve to define factor rings. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In the various branches of Mathematics that fall under the heading of Abstract algebra, the kernel of a Homomorphism measures the degree to which the homomorphism In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the Basic facts about ideals, homomorphisms and factor rings are recorded in the isomorphism theorems and in the Chinese remainder theorem. In Mathematics, the isomorphism theorems are three Theorems applied widely in the realm of Universal algebra, stating the existence of certain Natural The Chinese remainder theorem is a result about congruences in Number theory and its generalizations in Abstract algebra.

A ring is called commutative if its multiplication is commutative. In Mathematics, commutativity is the ability to change the order of something without changing the end result Commutative rings resemble familiar number systems, and various definitions for commutative rings are designed to recover properties known from the integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Commutative rings are also important in algebraic geometry. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with In commutative ring theory, numbers are often replaced by ideals, and the definition of prime ideal tries to capture the essence of prime numbers. In Mathematics, a prime ideal is a Subset of a ring which shares many important properties of a Prime number in the Ring of integers In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 Integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility. In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers. In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i Euclidean domains are integral domains in which the Euclidean algorithm can be carried out. In Abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies In Mathematics, the greatest common divisor (gcd, sometimes known as the greatest common factor (gcf or highest common factor (hcf, of two non-zero Important examples of commutative rings can be constructed as rings of polynomials and their factor rings. In Mathematics, a polynomial is an expression constructed from Variables (also known as indeterminates and Constants using the operations Summary: Euclidean domain => principal ideal domain => unique factorization domain => integral domain => Commutative ring. In Abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm applies In Abstract algebra, a principal ideal domain, or PID is an Integral domain in which every ideal is principal i In Mathematics, a unique factorization domain (UFD is roughly speaking a Commutative ring in which every element with special exceptions can be uniquely written In Abstract algebra, a branch of Mathematics, an integral domain is a Commutative ring with an additive identity 0 and a multiplicative identity 1 such In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property

Non-commutative rings resemble rings of matrices in many respects. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally Following the model of algebraic geometry, attempts have been made recently at defining non-commutative geometry based on non-commutative rings. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with Noncommutative geometry, or NCG, is a branch of Mathematics concerned with the possible spatial interpretations of Algebraic structures for which the Non-commutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets A module over a ring is an Abelian group that the ring acts on as a ring of endomorphisms, very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces. In Abstract algebra, the concept of a module over a ring is a generalization of the notion of Vector space, where instead of requiring the scalars In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division Examples of non-commutative rings are given by rings of square matrices or more generally by rings of endomorphisms of Abelian groups or modules, and by monoid rings. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Abstract algebra, a monoid ring is a new ring constructed from some other ring and a Monoid.

Some useful theorems

Generalizations

Any ring can be seen as a preadditive category with a single object. In Abstract algebra, the Artin–Wedderburn theorem is a Classification theorem for semisimple rings. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category It is therefore natural to consider arbitrary preadditive categories to be generalizations of rings. And indeed, many definitions and theorems originally given for rings can be translated to this more general context. Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and

References

Dictionary

ring theory

-noun

  1. The branch of mathematics dealing with the algebraic structure of rings.
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