Ring theory

In abstract algebra, 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. 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.

Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory. Because these two fields are so intimately connected with commutative rings, their theories are usually considered to be part of commutative algebra and field theory rather than of general ring theory.

Noncommutative rings are quite different in flavour, since more unusual behavior can arise. While the theory has developed in its own right, a fairly recent trend has sought to parallel the commutative development by building the theory of certain classes of noncommutative rings in a geometric fashion as if they were rings of functions on (non-existent) 'noncommutative spaces'. This trend started in the 1980s with the development of noncommutative geometry and with the discovery of quantum groups. It has led to a better understanding of noncommutative rings, especially noncommutative Noetherian rings. (Goodearl 1989)

Please refer to the glossary of ring theory for the definitions of terms used throughout ring theory.

Contents

History

Commutative ring theory originated in algebraic number theory, algebraic geometry, and invariant theory. Central to the development of these subjects were the rings of integers in algebraic number fields and algebraic function fields, and the rings of polynomials in two or more variables. Noncommutative ring theory began with attempts to extend the complex numbers to various hypercomplex number systems. The genesis of the theories of commutative and noncommutative rings dates back to the early 19th century, while their maturity was achieved only in the third decade of the 20th century.

More precisely, William Rowan Hamilton put forth the quaternions and biquaternions; James Cockle presented tessarines and coquaternions; and William Kingdon Clifford was an enthusiast of split-biquaternions, which he called algebraic motors. These non-commutative algebras, and the non-associative Lie algebras, were studied within universal algebra before the subject was divided into particular mathematical structure types. One sign of re-organization was the use of direct sums to describe algebraic structure.

The various hypercomplex numbers were identified with matrix rings by Joseph Wedderburn (1908) and Emil Artin (1928). Wedderburn's structure theorems were formulated for finite-dimensional algebras over a field while Artin generalized them to Artinian rings.

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%2Bc) = (a*b) %2B (a*c)
(a%2Bb) * c = (a*c) %2B (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. The number 1 is a common example of a unity.

The ring in which e is equal to the additive identity must have only one element. This ring is called the trivial ring.

Rings that sit inside other rings are called subrings. Maps between rings which respect the ring operations are called ring homomorphisms. Rings, together with ring homomorphisms, form a category (the category of rings). Closely related is the notion of ideals, certain subsets of rings which arise as kernels of homomorphisms and can serve to define factor rings. Basic facts about ideals, homomorphisms and factor rings are recorded in the isomorphism theorems and in the Chinese remainder theorem.

A ring is called commutative if its multiplication is commutative. Commutative rings resemble familiar number systems, and various definitions for commutative rings are designed to recover properties known from the integers. Commutative rings are also important in algebraic geometry. In commutative ring theory, numbers are often replaced by ideals, and the definition of prime ideal tries to capture the essence of prime numbers. 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. Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers. Euclidean domains are integral domains in which the Euclidean algorithm can be carried out. Important examples of commutative rings can be constructed as rings of polynomials and their factor rings. Summary: Euclidean domain => principal ideal domain => unique factorization domain => integral domain => Commutative ring.

Non-commutative rings resemble rings of matrices in many respects. Following the model of algebraic geometry, attempts have been made recently at defining non-commutative geometry based on non-commutative rings. Non-commutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules. 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. 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.

Some useful theorems

General:

Structure theorems:

Generalizations

Any ring can be seen as a preadditive category with a single object. 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.

References