Ring (mathematics)

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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, which itself is an example of a ring. Other examples include the polynomials and the integers modulo n. The branch of abstract algebra which studies rings is called ring theory.

Contents

1 Formal definition

1.1 Alternative definitions

2 Examples
3 Basic theorems
4 Constructing new rings from given ones
5 Categorical description
6 See also

Formal definition

A ring is a set R equipped with two binary operations + and ·, called addition and multiplication, such that:

(R, +) is an abelian group with identity element 0:
* (a + b) + c = a + (b + c)
* 0 + a = a + 0 = a
* a + b = b + a
* For every a in R, there exists an element denoted −a, such that a + −a = −a + a = 0
(R, ·) is a monoid with identity element 1:
* (a·b)·c = a·(b·c)
* 1·a = a·1 = a
Multiplication distributes over addition:
* a·(b + c) = (a·b) + (a·c)
* (a + b)·c = (a·c) + (b·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).

Although ring addition is commutative, such 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. Not all rings are commutative. For example [M_n(K)], the ring of [n\times n] matrices over a field K, is a non-commutative ring (n>1).

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, such that if there is an element b in the ring such that a·b = b·a = 1. If that is the case, 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).

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, 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 identity. Authors such as Bourbaki, who do require rings to have an identity, call algebraic objects which meet all the requirements of a ring except the identity requirement pseudo-rings.

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

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

Examples

The trivial ring has only one element which serves both as additive and multiplicative identity. Note that some authors define the ring as to specifically exclude this from being considered a ring.
The motivating example is the ring of integers with the two operations of addition and multiplication. This is a commutative ring.
* The rational, real and complex numbers form rings (in fact, they are even fields). These are likewise commutative rings.
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.
* The set of formal power series RX₁,...,X_n over a commutative ring R is a ring.
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).
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).
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 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 invertible with respect to addition). For instance, there is no natural number which can be added to 3 to get 0 as a result. To make it a ring one needs to add 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).

Basic theorems

From the axioms, one can immediately deduce that, for all elements a and b of a ring, we have

0a = a0 = 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

The identity element 1 is unique
If the ring has at least two elements then 0 ≠ 1
The binomial theorem [(x+y)^n=\sum_^nx^ky^] works in a ring if xy=yx

Constructing new rings from given ones

For every ring R we can define the opposite ring R^op by reversing the muliplication 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.
If a subset S of a ring R is closed under multiplication and subtraction and contains the multiplicative identity element, 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 (R_j)_jεJ, the direct product and direct sum exist. The direct product is the collection of "infinite-tuples" (r_j)_jεJ 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 jεJ is the set of all maps r:J→U with the property that r_jεR_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.

The direct sum of a collection of rings (R_j)_jεJ is the subring of the direct product consisting of all infinite-tuples (r_j)_jεJ with the property that r_j=0 for all but finitely many j. In particular, if J is finite, then the direct sum and the direct product are isomorphic, but in general they have quite different properties.

Given a ring R and an 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.

Categorical description

Just as monoids and groups can be viewed as categories with a single object, rings can be viewed as (pre)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 in ring addition. The opposite ring is then the categorical dual.