In mathematics, a ring is an algebraic structure which generalizes the algebraic properties of the integers. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and 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 Rings, unlike groups, contain two operations usually called addition and multiplication. 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 The branch of abstract algebra which studies rings is called ring theory. Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those

Contents 1 Motivation 2 Formal definition 2.1 Alternative definitions 3 Examples 4 Basic theorems 5 Constructing new rings from given ones 6 Categorical description 7 See also 8 References

Motivation

In mathematics, objects commonly arise which have structure similar to the integers, but may behave differently in some ways. For example, matrices can be added and multiplied as expected, but such multiplication does not in general satisfy the commutative law. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, commutativity is the ability to change the order of something without changing the end result As a different example, the integers modulo n satisfy similar laws of arithmetic but have zero divisors if n is not prime. The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure In Abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0

A ring is an abstraction of certain properties of the integers that is general enough to allow the study of a greater variety of objects, but strong enough to ensure a rich theory in which substantial results can be proven. In a sense, rings have more structure than an abelian group but less than a field. 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 Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division That is, every ring is an abelian group and every field is a ring.

Formal definition

A ring is a set R equipped with two binary operations + : R × R → R and ⋅ : R × R → R (where × denotes the Cartesian product), called addition and multiplication, such that:

• (R, +) is an abelian group with identity element 0, meaning that for all a, b, c in R, the following axioms hold:
  • (a + b) + c = a + (b + c) (+ is associative)
  • 0 + a = a (0 is the identity)
  • a + b = b + a (+ is commutative)
  • for each a in R there exists −a in R such that a + (−a) = (−a) + a = 0 (−a is the inverse element of a)
• (R, ⋅) is a monoid with identity element 1, meaning that for all a, b, c in R, the following axioms hold:
  • (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) (⋅ is associative)
  • 1 ⋅ a = a ⋅ 1 = a (1 is the identity)
• Multiplication distributes over addition:
  • a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c)
  • (a + b) ⋅ c = (a ⋅ c) + (b ⋅ c). In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory. 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, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, associativity is a property that a Binary operation can have In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law

As with groups the symbol ⋅ is usually omitted and multiplication is just denoted by juxtaposition. 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 Also, the standard order of operation rules are used, so that, for example, a + bc is an abbreviation for a + (b ⋅ c). In Algebra and Computer programming, when a number or expression is both preceded and followed by a Binary operation, a rule is required for which operation

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. In Mathematics, commutativity is the ability to change the order of something without changing the end result Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property An example of a non-commutative ring is the ring of n × n matrices over a field K, for n > 1. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

Rings need not have multiplicative inverses either. In Mathematics, a multiplicative inverse for a number x, denoted by 1&frasl x or x &minus1 is a number which 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. In Mathematics, a unit in a ( Unital) ring R is an invertible element of R, i The set of all units in R forms a group under ring multiplication; this group is denoted by U(R) or R*. 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

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 excludes only one ring: the so called trivial ring or zero ring, which has only a single element. A trivial ring is a ring defined on a Singleton set, { r } The ring operations (× and + are trivial r \times r = r
• A more significant difference is that some authors omit the requirement that a ring have a multiplicative identity. ^[1][2] These authors call rings which do have multiplicative identities unital rings, unitary rings, or simply rings with unity or rings with identity. 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. Nicolas Bourbaki is the collective Pseudonym under which a group of (mainly French) 20th-century Mathematicians wrote a series of books presenting an exposition The term rng (jocular; ring without the multiplicative identity) has also been used. In Abstract algebra, a rng (also called a pseudo-ring or non-unital ring) is an Algebraic structure satisfying the same properties as a The even integers are an example of a pseudo-ring. Any non-unitary ring R can be embedded in a canonical way as a sub-pseudo-ring of a unitary ring, namely R ⊕ Z with multiplication defined by (x, m) ⋅ (y, n) = (xy + my + nx, mn), so that (0, 1) is a multiplicative identity. 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. (See Unital. In Mathematics, an algebra is unital (some authors say unitary) if it contains a multiplicative Identity element (or unit) i )
• 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. In Mathematics, associativity is a property that a Binary operation can have See nonassociative rings for a discussion of the more general situation. In Abstract algebra, a nonassociative ring is a generalization of the concept of ring.

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. Commutative algebra is the branch of Abstract algebra that studies Commutative rings their ideals, and modules over such rings Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with 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". Experimental infobox see Wikipedia talkPersondata before changing --> Alexander Grothendieck (born March 28, 1928 in Berlin, Germany The Éléments de géométrie algébrique ("Elements of Algebraic Geometry " by Alexander Grothendieck (assisted by Jean Dieudonné

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

Examples

• The trivial ring {0} has only one element, and it serves both as the additive and the multiplicative identity.
• The motivating example is the ring of integers with the two operations of addition and multiplication. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French This is a commutative ring. In Mathematics, commutativity is the ability to change the order of something without changing the end result
  • The rational, real and complex numbers form rings (in fact, they are even fields). In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division These are likewise commutative rings.
• Every field is by definition a commutative ring.
• The Gaussian integers form a ring, as do the Eisenstein integers. A Gaussian integer is a Complex number whose real and imaginary part are both Integers The Gaussian integers with ordinary addition and multiplication of complex In Mathematics, Eisenstein integers, named after Ferdinand Eisenstein, are Complex numbers of the form z = a + b\omega \\!
• The polynomial ring R[X] of polynomials over a ring R is also a ring. 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
  • The set of formal power series R[[X₁, …, X_n]] over a commutative ring R is 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
• 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. In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally For n=1, this matrix ring is just (isomorphic to) R itself. For n>1, 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 Z_n = 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). In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers If n=1, then Z/nZ is the trivial ring.
• 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. In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) In Mathematics, the symmetric difference of two sets is the set of elements which are in one of the sets 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 This is an example of a Boolean ring. In Mathematics, a Boolean ring R is a ring (with identity for which x 2 = x for all x in R; that
• The set of all continuous real-valued functions defined on the interval [a, b] forms a ring (even an associative algebra). In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive 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. 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, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function 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. 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, a group ring is a ring R constructed from a ring R and a group G (written multiplicatively In Mathematics, a free module is a Free object in the category of modules Given a set S, a free module on S is a (particular construction 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). In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an 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, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to Addition is the mathematical process of putting things together For instance, there is no natural number which can be added to 3 to get 0 as a result. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an There is a natural way to make it a ring by adding negative numbers to the set, thus obtaining the ring of integers. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French The natural numbers (including 0) form an algebraic structure known as a semiring (which has all of the properties of a ring except the additive inverse property). In Abstract algebra, a semiring is an Algebraic structure similar to a ring, but without the requirement that each element must have an Additive inverse
• 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.
• Ring of dual numbers: Let є be a formal symbol and F a field. A variety of dualities in mathematics are listed at Duality (mathematics. The ring of dual numbers, F[є], is defined as F[є] = {a + bє : a, b in F},with the following addition and multiplication:
  (a + bє) + (c + dє) = a + c + (b + d)є
  (a + bє)(c + dє) = ac + (ad + bc)є
  Note that є is a zero divisor: є ≠ 0 but є² = 0.
• Ring of split-complex numbers: z = x + y j , j² = +1. In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real A ring analogous to the ordinary complex plane but substitutes conjugate hyperbolas for the unit circle.

Basic theorems

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

• 0 ⋅ a = a ⋅ 0 = 0. This is known as the "dominative property"
• (−1)a = −a
• (−a)b = a(−b) = −(ab)
• (ab)⁻¹ = b⁻¹a⁻¹ if both a and b are invertible. In Mathematics, the idea of inverse element generalises the concepts of negation, in relation to Addition, and reciprocal, in relation to

Other basic theorems

• The identity element 1 is unique. 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
• 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

holds whenever x and y commute. In Mathematics, the binomial theorem is an important Formula giving the expansion of powers of Sums Its simplest version says The theorem holds for arbitrary x and y in a commutative ring. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property

• If a ring is a cyclic group under addition, then it is commutative.

Constructing new rings from given ones

• For every ring R we can define the opposite ring R^op by reversing the multiplication in R. Given the multiplication · in R the multiplication ∗ in R^op is defined as b ∗ a := a ⋅ b. The "identity map" from R to R^op is an isomorphism if and only if R is commutative. However, even if R is not commutative, it is still possible for R and R^op to be isomorphic. For example, if R is the ring of n × n matrices of real numbers, then the transposition map from R to R^op is an isomorphism. This article is about the Matrix Transpose operator For other uses see Transposition In Linear algebra, the transpose of a
• 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. 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
• 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 term center or centre is used in various contexts in Abstract algebra to denote the set of all those elements that commute with all other elements 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). In Mathematics, it is possible to combine several rings into one large product ring. Cartesian square redirects here For Cartesian squares in Category theory, see Cartesian square (category theory.

• More generally, for any index set J and collection of rings (R_j)_{j ∈ J}, there is a direct product ring. In Mathematics, it is possible to combine several rings into one large product ring. The direct product is the collection of "infinite-tuples" (r_j)_{j ∈ J} with component-wise addition and multiplication. More formally, let U be the union of all of the rings R_j. Then the direct product of the R_j over all j ∈ J is the set of all maps r : J → U with the property that r_j ∈ R_j. Addition and multiplication of these functions is via the addition and multiplication in each individual R_j. Thus

(r + s)_j = r_j + s_j and (rs)_j = r_js_j.

• 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. In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the

• 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 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, the Tensor product of two ''R''-algebras is also an R -algebra in a natural way

Categorical description

Rings can be thought of as monoids in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product). In Category theory, a monoid (or monoid object) (M\mu\eta in a Monoidal category C is an object M together with two In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype In Mathematics, a monoidal category (or tensor category) is a category C equipped with a Bifunctor &otimes: C In Mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (roughly speaking "multiplication" to be carried out in The monoid action of a ring R on a abelian group is simply an R-module. 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

It follows that a ring may be regarded as a preadditive category (a category enriched over Ab) with a single object. In Mathematics, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Category theory and its applications to Mathematics, an enriched category is a category whose Hom-sets are replaced by objects from some other Here the morphisms are the ring elements, composition of morphisms is ring multiplication, and the additive structure on morphisms is ring addition. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and The opposite ring is then the categorical dual. In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the

See also

References

1. ^ Herstein, I. N. Topics in Algebra, Wiley; 2 edition (June 20, 1975), ISBN 0-471-01090-1. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those Ring theory is the branch of Mathematics in which rings are studied that is structures supporting both an Addition and a Multiplication operation In Mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity and whose Morphisms In Mathematics, specifically in Ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the In Abstract algebra, a nonassociative ring is a generalization of the concept of ring. In Algebra, a branch of Pure mathematics, an algebraic structure consists of one or more sets closed under one or more operations, In Mathematics, a Boolean ring R is a ring (with identity for which x 2 = x for all x in R; that In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property In Abstract algebra, an ordered ring is a Commutative ring R with a Total order \leq such that if a\leq In Mathematics, differential rings differential fields and differential algebras are rings, fields and algebras equipped with a derivation, In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division 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 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 Israel Nathan Herstein ( March 28, 1923, Lublin, Poland – February 9, 1988, Chicago, Illinois) was
2. ^ Joseph Gallian (2004). Contemporary Abstract Algebra. Houghton Mifflin. ISBN 9780618514717.  

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