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. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In universal algebra, a branch of pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, satisfying some axioms. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

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. ...

History

The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. Furthermore, the appearance of hypercomplex numbers in the mid-nineteenth century undercut the pre-eminence of fields in mathematical analysis. In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, an algebraic integer is a complex number α that is a root of an equation P(x) = 0 where P(x) is a monic polynomial (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients. ... In mathematics, hypercomplex numbers are extensions of the complex numbers constructed by means of abstract algebra, such as quaternions, tessarines, coquaternions, octonions, biquaternions and sedenions. ... This article presents the essential definitions. ...

Richard Dedekind introduced the concept of a ring. Richard Dedekind Julius Wilhelm Richard Dedekind (October 6, 1831 – February 12, 1916) was a German mathematician who did important work in abstract algebra and the foundations of the real numbers. ...

The term ring (Zahlring) was coined by David Hilbert in the article Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 4, 1897. David Hilbert David Hilbert (January 23, 1862, Wehlau, East Prussia–February 14, 1943, Göttingen, Germany) was a German mathematician, recognized as one of the most influential mathematicians of the 19th and early 20th centuries. ...

The first axiomatic definition of a ring was given by Adolf Fraenkel in an essay in Journal für die reine und angewandte Mathematik (A. L. Crelle), vol. 145, 1914. Adolf Abraham Halevi Fraenkel (February 17, 1891 - October 15, German / Israeli mathematician. ... Crelles Journal, or just Crelle, is the common name for the Journal für die reine und angewandte Mathematik founded by August Leopold Crelle. ...

In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper Ideal Theory in Rings. Emmy Noether (born Nöther) (March 23, 1882 – April 14, 1935) was a talented mathematician of the early 20th century, with penetrating insights that she used to develop elegant abstractions which she formalized and published. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

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, In mathematics, an abelian group, also called a commutative group, is a group such that for all a and b in G. In other words, the order of elements in a product doesnt matter. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ...

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

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

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

and such that there exists a multiplicative identity, or unity, that is, an element 1 so that for all a in R,

a * 1 = 1 * a = a

It is simple to show that any ring in which 1 = 0 must have just one element; any such ring is called a zero 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. 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. In abstract algebra, a branch of mathematics, a subring is a subset of a ring containing the multiplicative identity, which is itself a ring under the same binary operations. ... In abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... 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 fails to be injective. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, the isomorphism theorems are three theorems, applied widely in the realm of universal algebra, stating the existence of certain natural isomorphisms. ... link titleThe Chinese remainder theorem (CRT) is the name for several related results in abstract algebra and number theory. ...

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. In mathematics, especially abstract algebra, a binary operation * on a set S is commutative if x * y = y * x for all x and y in S. Otherwise * is noncommutative. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... 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 that has exactly two (distinct) natural number divisors, which are 1 and the prime number itself. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ... In mathematics, the greatest common divisor (gcd), sometimes known as the greatest common factor (gcf) or highest common factor (hcf) of two integers which are both not zero is the largest integer that divides both numbers. ... In mathematics, a polynomial is an expression in which constants and variables are combined using only addition, subtraction, multiplication, and positive whole number exponents (raising to a power). ... In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. ... In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In mathematics, a unique factorization domain (UFD) is, roughly speaking, a commutative ring in which every element can be uniquely written as a product of prime elements, analogous to the fundamental theorem of arithmetic for the integers. ... In abstract algebra, an integral domain is a commutative ring with an additive identity 0 and a multiplicative identity 1 such that 0 ≠ 1, in which the product of any two non-zero elements is always non-zero; that is, there are no zero divisors. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

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. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... 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 to lie in a field, the scalars may lie in an arbitrary ring. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In mathematics, an endomorphism is a morphism (or homomorphism) from a mathematical object to itself. ... This article presents the essential definitions. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In abstract algebra, a monoid ring is a procedure which constructs a new ring from a given ring and a monoid. ...

Some useful theorems

• Artin-Wedderburn theorem

In abstract algebra, the Artin-Wedderburn theorem is a classification theorem for semisimple product of ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely determined up to permutation of the index i. ...

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. A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ...

References

• History of ring theory at the MacTutor Archive
• R.B.J.T. Allenby (1991). Rings, Fields and Groups. Butterworth-Heinemann. ISBN 0-3405-4440-6.
• T.S. Blyth and E.F. Robertson (1985). Groups, rings and fields: Algebra through practice, Book 3. Cambridge university Press. ISBN 0-521-27288-2.