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Encyclopedia > Ring (mathematics)

In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. A ring is a generalization of the integers. Other examples include the polynomials and the integers modulo n. The branch of abstract algebra which studies rings is called ring theory. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... 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 are commonly denoted by the above symbol. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... The word modulo (Latin, with respect to a modulus of ___) is the Latin ablative of modulus which itself means a small measure. ... Abstract algebra is the field of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras. ... 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. ...

## Contents

A ring is a set $,R$ equipped with two binary operations $+colon R rightarrow R times R$ and $cdotcolon R rightarrow R times R$ (where $times$ denotes the Cartesian product), called addition and multiplication, such that: In mathematics, a set can be thought of as any collection of distinct objects considered as a whole. ... In mathematics, a binary operation is a calculation involving two input quantities, in other words, an operation whose arity is two. ... In mathematics, the Cartesian product is a direct product of sets. ...

• $,(R, +)$ is an abelian group with identity element $,0$, so that $,forall a, b, c in R$, the following axioms hold:
• $,a + b in R$
• $,(a + b) + c = a + (b + c)$
• $,0 + a = a + 0 = a$
• $,a + b = b + a$
• $,exists$ $,-a in R$ such that $,a + (-a) = (-a) + a = 0$
• $,(R, cdot)$ is a monoid with identity element $,1$, so that $,forall a, b, c in R$, the following axioms hold:
• $,a cdot b in R$
• $,(a cdot b)cdot c = acdot(b cdot c)$
• $,1cdot a = acdot 1 = a$
• $,a cdot (b + c) = (a cdot b) + (a cdot c)$
• $,(a + b)cdot c = (a cdot c) + (b cdot c)$

As with groups the symbol · is usually omitted and multiplication is just denoted by juxtaposition. Also, the standard order of operation rules are used, so that, for example, a+bc is an abbreviation for a+(b·c). 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, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalises the distributive law from elementary algebra. ... This picture illustrates how the hours in a clock form a group. ... In arithmetic and algebra, when a number or expression is both preceded and followed by a binary operation, a rule is required for which operation should be applied first. ...

Although ring addition is commutative, so that a+b = b+a, ring multiplication is not required to be commutative; a·b need not equal b·a. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. An example of a non-commutative ring is the ring of n×n matrices over a field K, for n > 1. 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 are commonly denoted by the above symbol. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

Rings need not have multiplicative inverses either. An element a in a ring is called a unit if it is invertible with respect to multiplication: if there is an element b in the ring such that a·b = b·a = 1, then b is uniquely determined by a and we write a−1 = b. The set of all units in R forms a group under ring multiplication; this group is denoted by U(R) or R*. The reciprocal function: y = 1/x. ... In mathematics, a unit in a ring R is an element u such that there is v in R with uv = vu = 1R. That is, u is an invertible element of the multiplicative monoid of R. The units of R form a group U(R) under multiplication, the group of... This picture illustrates how the hours in a clock form a group. ...

### Alternative definitions

There are some alternative definitions of rings of which the reader should be aware:

• Some authors add the additional requirement that 0 ≠ 1. This omits only one ring: the so called trivial ring or zero ring, which has only a single element.
• A more significant difference is that some authors (such as I. N. Herstein) omit the requirement that a ring have a multiplicative identity. These authors call rings which do have multiplicative identities unital rings, unitary rings, or simply rings with unity. Authors such as Bourbaki, who do require rings to have a multiplicative identity, call algebraic objects which meet all the requirements of a ring except possibly the unity requirement pseudo-rings. The term rng (jocular; ring without the multiplicative identity) has also been used.

Any non-unitary ring R can be embedded in a canonical way as a subrng of a unitary ring, namely RZ with (0,1) as unit element and multiplication defined in the expected manner. This process is said to adjoin a unit element to R. If the same construction of adjoining a unit is applied to unitary ring R, the result is a different ring, with a new unit element. A trivial ring is a ring defined on a singleton set, {x}. The ring operations (* and +) are trivial: x * x = x x + x = x Clearly this ring is commutative. ... Israel Nathan Herstein (1923-1988) was a mathematician, appointed as professor at the University of Chicago in 1951. ... Nicolas Bourbaki is the collective allonym under which a group of (mainly French) 20th-century mathematicians wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935. ... In abstract algebra a rng is an algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element. ...

• Similarly, the requirement for the ring multiplication to be associative is sometimes dropped, and rings in which the associative law holds are then called associative rings. See nonassociative rings for a discussion of the more general situation.

As noted above, multiplication in a ring need not be commutative. Some fields such as commutative algebra and algebraic geometry are primarily concerned with commutative rings. Mathematicians writing in those areas (such as Alexander Grothendieck in Éléments de géométrie algébrique) frequently use the word ring to mean "commutative ring" by convention, and not necessarily commutative ring to mean "ring". In mathematics, associativity is a property that a binary operation can have. ... In abstract algebra, a nonassociative ring is a generalization of the concept of ring. ... In abstract algebra, commutative algebra studies commutative rings, their ideals, and modules over such rings. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Alexander Grothendieck (born March 28, 1928 in Berlin, Germany) is one of the most important mathematicians of the 20th century. ... The Ã‰lÃ©ments de gÃ©omÃ©trie algÃ©brique (Elements of Algebraic Geometry) by Alexander Grothendieck (assisted by Jean DieudonnÃ©), or EGA for short, are an unfinished 1500-page treatise, in French, on algebraic geometry that was published (in eight parts or fascicles) from 1960 through 1967 by the Institut...

In this article all rings are assumed to be associative and unital unless otherwise stated.

## Examples

• The trivial ring {0} has only one element which serves both as additive and multiplicative identity.
• The motivating example is the ring of integers with the two operations of addition and multiplication. This is a commutative ring.
• Every field is by definition a commutative ring.
• The Gaussian integers form a ring, as do the Eisenstein integers.
• The polynomial ring R[X] of polynomials over a ring R is also a ring.
• Example of a noncommutative ring: For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations. For n=1, this matrix ring is just (isomorphic to) R itself. For n>2, this matrix ring is an example of a noncommutative ring (unless R is the trivial ring).
• Example of a finite ring: If n is a positive integer, then the set Zn = Z/nZ of integers modulo n (as an additive group the cyclic group of order n) forms a ring with n elements (see modular arithmetic).
• If S is a set, then the power set of S becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring.
• The set of all continuous real-valued functions defined on the interval [a, b] forms a ring (even an associative algebra). The operations are addition and multiplication of functions.
• If G is an abelian group, then the endomorphisms of G form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms.
• If G is a group and R is a ring, the group ring of G over R is a free module over R having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G.
• Non-example: The set of natural numbers N is not a ring, since (N, +) is not even a group (the elements are not all invertible with respect to addition). For instance, there is no natural number which can be added to 3 to get 0 as a result. There is a natural way to make it a ring by adding negative numbers to the set, thus obtaining the ring of integers. The natural numbers form an algebraic structure known as a semiring (which has all of the properties of a ring except the additive inverse property).
• The even numbers 2Z (including negative even numbers) are an example of a pseudo-ring in that they have all the properties of a ring except a multiplicative identity.

The integers are commonly denoted by the above symbol. ... 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. ... In mathematics, a rational number is a number which can be expressed as a ratio of two integers. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = âˆ’1. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... A Gaussian integer is a complex number whose real and imaginary part are both integers. ... Eisenstein integers as intersection points of a triangular lattice in the complex plane In mathematics, Eisenstein integers, named after Ferdinand Eisenstein, are complex numbers of the form where and a and b are integers and is a complex cube root of unity. ... In abstract algebra, a polynomial ring is the set of polynomials in one or more variables with coefficients in a ring. ... In mathematics, formal power series are devices that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of convergence. They are also useful to compactly describe sequences and to find closed formulas for recursively defined sequences; this is... 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 group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element a (called a generator of the group) such that, when written multiplicatively, every element of the group is a power of a (or na... Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers wrap around after they reach a certain value â€” the modulus. ... In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, the symmetric difference of two sets is the set of elements which are in one of either set, but not in both. ... In mathematics, the intersection of two sets A and B is the set that contains all elements of A that also belong to B (or equivalently, all elements of B that also belong to A), but no other elements. ... In mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists of idempotent elements. ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Graph of example function, The mathematical concept of a function expresses the intuitive idea of deterministic dependence between two quantities, one of which is viewed as primary (the independent variable, argument of the function, or its input) and the other as secondary (the value of the function, or output). A... In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... 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. ... This picture illustrates how the hours in a clock form a group. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... This picture illustrates how the hours in a clock form a group. ... In mathematics, the group ring is an algebraic construction that associates to a group G and a commutative ring with unity R an R-algebra R[G] (or sometimes just RG) such that the multiplication in R[G] is induced by the multiplication in G. R[G] can be described... In mathematics, a free module is a module having a free basis. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... This picture illustrates how the hours in a clock form a group. ... In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] Addition is the mathematical operation of combining or adding two numbers to obtain an equal simple amount or total. ... In mathematics, a natural number can mean either an element of the set {1, 2, 3, ...} (i. ... The integers are commonly denoted by the above symbol. ... In abstract algebra, a semiring is an algebraic structure, similar to a ring, but without additive inverses. ...

## Basic theorems

From the axioms, one can immediately deduce that if $,R$ is a ring, $,forall a, b in R$ we have:

• $,0 cdot a = a cdot 0 = 0.$
• $,(-1)a = -a$
• $,(-a)b = a(-b) = -ab$
• $,(ab)^{-1} = b^{-1} a^{-1}$ if both $,a$ and $,b$ are invertible.

Other basic theorems In mathematics, the idea of inverse element generalises both of the concepts of negation, in relation to addition (see additive inverse), and reciprocal, in relation to multiplication. ...

• The identity element 1 is unique.
• If a ring element has a multiplicative inverse, then the inverse is unique.
• If the ring has at least two elements then 0 ≠ 1
• If n is an integer, and a an element of the ring define na as one would by viewing a as an element of the additive group of the ring (that is, 0 if n is 0, the sum of n copies of a if n is positive, and the opposite of (–n)a if n is negative.) We usually write n for the ring element n1. Then:
• The two definitions of na coincide, that is, first, with n viewed as an integer as above; second, with n meaning the ring element n1 and multiplication in the expression na taking place in the ring. Thus the integer n may be identified with the ring element n. (Except that more than one integer may correspond to a single ring element this way.)
• The ring element n commutes with all other elements of the ring.
• If m and n are integers and a and b are ring elements, then (m·a)(n·b) = (mn)·(ab)
• If n is an integer and a is a ring element, then n·(-a) = -(n·a)
• The binomial theorem
$(x+y)^n=sum_{k=0}^n{n choose k}x^ky^{n-k},$
holds whenever x and y commute. This is true in any commutative ring.
• If a ring is a cyclic group under addition, then it is commutative.

In mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. ... In mathematics, the binomial theorem is an important formula giving the expansion of powers of sums. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ...

## Constructing new rings from given ones

• For every ring R we can define the opposite ring Rop by reversing the multiplication in R. Given the multiplication · in R the multiplication ∗ in Rop is defined as ba := a·b. The "identity map" from R to Rop is an isomorphism if and only if R is commutative. However, even if R is not commutative, it is still possible for R and Rop to be isomorphic. For example, if R is the ring of n×n matrices of real numbers, then the transposition map from R to Rop is an isomorphism.
• If a subset S of a ring R is closed under multiplication, addition and subtraction and contains the additive and multiplicative identity elements, then S is called a subring of R.
• The center of a ring R is the set of elements of R that commute with every element of R; that is, c lies in the center if cr=rc for every r in R. The center is a subring of R. We say that a subring S of R is central if it is a subring of the center of R.
• The direct product of two rings R and S is the cartesian product R×S together with the operations
(r1, s1) + (r2, s2) = (r1+r2, s1+s2) and
(r1, s1)(r2, s2) = (r1r2, s1s2).
• More generally, for any index set J and collection of rings (Rj)jεJ, there is a direct product ring. The direct product is the collection of "infinite-tuples" (rj)jεJ with component-wise addition and multiplication. More formally, let U be the union of all of the rings Rj. Then the direct product of the Rj over all jεJ is the set of all maps r:JU with the property that rjεRj. Addition and multiplication of these functions is via the addition and multiplication in each individual Rj. Thus
(r+s)j=rj+sj and (rs)j=rjsj.
• Given a ring R and an two-sided ideal I of R, the quotient ring (or factor ring) R/I is the set of cosets of I together with the operations
(a+I) + (b+I) = (a+b) + I and
(a+I)(b+I) = (ab) + I.
• Since any ring is both a left and right module over itself, it is possible to construct the tensor product of R over a ring S with another ring T to get another ring provided S is a central subring of R and T.

In linear algebra, the transpose of a matrix A is another matrix AT (also written Atr, tA, or Aâ€²) created by any one of the following equivalent actions: write the rows of A as the columns of AT write the columns of A as the rows of AT reflect A... 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. ... The term center is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. ... In mathematics, it is possible to combine several rings into one large product ring. ... In mathematics, the Cartesian product is a direct product of sets. ... In mathematics, it is possible to combine several rings into one large product ring. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... 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, the tensor product of two R-algebras is also an R-algebra in a natural way. ...

## Categorical description

Just as monoids and groups can be viewed as categories with a single object, rings can be viewed as additive categories with a single object. Here the morphisms are the ring elements, composition of morphisms is ring multiplication, and the additive structure on morphisms is ring addition. The opposite ring is then the categorical dual. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... This picture illustrates how the hours in a clock form a group. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ... In mathematics, specifically in category theory, an additive category is a preadditive category C such that any finitely many objects A1,...,An of C have a biproduct A1 &#8853; ··· &#8853; An in C. (Recall that a category C is preadditive if all its morphism sets are Abelian groups and morphism... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...

Wikibooks has a book on the topic of
Abstract algebra/Rings, fields and modules
• Special types of rings:

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 Chapter 9: Mathematics (500 words) Known as a Kaleidocycle, or Flexahedron, the toy was invented years ago by a bored mathematics student. Now pinch that end closed, so the tabs are glued to the paper, holding the snake's tail firmly in his mouth, forming a ring. When the glue is dry, you can now start turning the ring inside-out, pushing the center up from the bottom, and the outside parts down.
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