In mathematics, more specifically in modern algebra, a ring is a set equipped with two binary operations – often referred to as addition and multiplication. Despite their name, these two operations are not the same as the natural operations of addition and multiplication defined on the integers; rather, they are a generalization of these familiar operations. In particular, the operations of addition and multiplication defined on a ring, satisfy certain conditions, known as axioms, that make the ring structurally similar to the integers. However, these conditions do not restrict the ring too much, in that a broad variety of mathematical objects are taken into account. These two ideas, when put together, give rise to the ring axioms which are discussed later in the article (see below).
Ring theory is the branch of mathematics which studies rings. Rings, being structurally similar to the integers in many ways, have algebraic properties similar to the integers. For example, a ring theoretic analogue of the fundamental theorem of arithmetic exists, as well as a generalization of the Euclidean algorithm and Bézout's identity. However, this is not to say that these results hold in the context of all rings. As the ring axioms are very general, often it is found that structurally, rings are no more similar to the integers further than the ring axioms. That is, apart from the ring axioms that are shared by both the ring, and the set of integers, there are not many other properties that both structures satisfy. It is therefore necessary to impose additional conditions on the structure of rings, that allow results such as the fundamental theorem of arithmetic to hold true. In general, many properties of the integers, relevant to only the operations of addition and multiplication, can be generalized to the context of rings (notice that subtraction and division are inverses of the operations of addition and multiplication respectively, and therefore can also be defined in the context of rings (under certain conditions)). However, ring theory studies rings in their full generality, and often the ring axioms are all that is necessary to develop a rich theory.
Although ring theory deals with only a single mathematical structure (rings), and therefore deals with only one structural aspect of the integers, it has far-reaching applications in mathematics. For example, algebraic geometry, the combination of ring theory with the language and problems of geometry, may be used to give a proof of Fermat's last theorem - a famous but now solved problem in mathematics. In a similar manner, many number theoretic facts may be proved using techniques of ring theory; sometimes there being no proof (mathematics) of a number theoretic fact without the use of ring theory. This is not to say, however, that ring theory only has applications in number theory. For example, ring theory has many important applications in general topology, and in particular, algebraic topology, where rings can serve as certain invariants of topological spaces - or, in more formal language, there exist certain functors between the category of topological spaces and the category of rings. One example being the cohomology ring, associated to any topological space. As another example of the applications of ring theory to mathematics, to any group is associated its Burnside ring which uses a ring to describe the various ways a group may act on a finite set. These examples provide only a taste of the applications of ring theory in mathematics - there are many other such applications to be discussed later in this article (see below).
Although ring theory is very much central and basic to much of mathematics, it is still a major area of research. A huge area of current reasearch in ring theory, lies within algebraic geometry and algebraic number theory. Ring theory alone, is also a major area of research - the main aspect of research being the distinction between commutative rings, and non-commmutative rings. Commutative rings are fairly well understood compared to non-commutative rings, due to the fact that a much smaller class of rings are to be considered. On the other hand, many results in non-commutative ring theory may be easily transferred to commutative ring theory, while retaining the same set of hypothesis. Despite these facts, there are many important open problems in both commutative and non-commutative ring theory.
The most familiar example of a ring is the set of all integers, Z, consisting of the numbers
together with the usual operations of addition and multiplication. These operations satisfy the following properties:
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. To qualify as a ring, the set and two operations, (R, +, · ), must satisfy the following requirements known as the ring axioms. [2]
As with some axiomatic theories, there are often disputes as to what axioms a ring should satisfy. Sometimes the disagreement between two definitions is minor. For instance, some authors insist that 1 ≠ 0 in a ring (in words, this means that the multiplicative identity of the ring must be different from its additive identity). In particular they don't consider the trivial ring to be a ring (see below).
A more significant disagreement is that some authors omit the existence of a multiplicative identity in a ring[3][4] [5] . For instance, this would allow the even integers to form a ring with the natural operations of addition and multiplication (all ring axioms are satisfied except for the existence of a multiplicative identity). Rings that satisfy the ring axioms as given above but do not contain a multiplicative identity are sometimes called pseudo-rings. The term rng (jocular; ring without the multiplicative identity) is also used for such rings. Rings which do have multiplicative identities (and also satisfy the above axioms) are sometimes referred to unital rings, unitary rings, rings with unity, rings with identity or rings with 1.[6] Note that one can always embed a non-unitary ring inside a unitary ring (see this for one particular construction of this embedding).
There are still other more significant differences between two particular definitions of a ring. For instance, some authors omit associativity of multiplication in the set of ring axioms; rings that are nonassociative are called nonassociative rings. In this article, all rings are assumed to satisfy the axioms as given above unless stated otherwise.
Consider the set Z4 consisting of the numbers 0, 1, 2, 3 where addition and multiplication are defined as follows (note that for any integer, x, x mod 4 is defined to be the remainder when x is divided by 4):
It is simple (but tedious) to verify that Z4 is a ring under these operations. First of all, one can use the left-most table to show that Z4 is closed under addition (any result is either 0, 1, 2 or 3). Associativity of addition in Z4 follows from associativity of addition in the set of all integers. The additive identity is 0 as can be verified by looking at the left-most table. Given an integer x, there is always an inverse of x; this inverse is given by 4 - x as one can verify from the additive table. Therefore, Z4 is an abelian group under addition.
Similarly, Z4 is closed under multiplication as the right-most table shows (any result above is either 0, 1, 2 or 3). Associativity of multiplication in Z4 follows from associativity of multiplication in the set of all integers. The multiplicative identity is 1 as can be verified by looking at the right-most table. Therefore, Z4 is a monoid under multiplication.
Distributivity of the two operations over each other follow from distributivity of addition over multiplication (and vice-versa) in Z (the set of all integers).
Therefore, this set does indeed form a ring under the given operations of addition and multiplication.
Properties of this ring
If we define on the singleton set {0}:
0 + 0 = 0
0 × 0 = 0
then one can verify that ({0}, +, ×) forms a ring known as the trivial ring. Since there can be only one result for any product or sum (0), this ring is both closed and associative for addition and multiplication, and furthermore satisfies the distributive law. The additive and multiplicative identities are both equal to 0. Similarly, the additive inverse of 0 is 0. The trivial ring is also a (rather trivial) example of a zero ring (see below).
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.
Richard Dedekind (image to the right) introduced the concept of a ring. [7]
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. [7]. The motivation for this term is unclear.
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. [7]
In 1921, Emmy Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper Ideal Theory in Rings.[7]
Basic facts about rings that can be deduced from the ring axioms are commonly subsumed under elementary ring theory. For example, one can deduce uniqueness of the additive identity in a ring, and uniqueness of the additive inverse of a particular element in a ring using group theory alone. Similarly, the inverse of a unit in a ring is necessarily unique (provable using group theory). However, some properties that effectively combine the additive and multiplicative structures in a ring cannot be proved by group theoretical means alone. For example, one cannot prove that for every element a in a ring (R, +, ⋅), a ⋅ 0 = 0 ⋅ a = 0, using only one of the structures. One also must take advantage of distributivity of multiplication over addition (see below). This constitutes one explanation as to why ring theory is important in its own right. This also explains why distributivity is such an important axiom in ring theory.
From the axioms, one can deduce that if (R, +, ⋅) is a ring, for all a, b in R we have:
Theorem 1: 0 ⋅ a = a ⋅ 0 = 0
0 ⋅ a = (0 + 0) ⋅ a = (0 ⋅ a) + (0 ⋅ a)
By subtracting 0 ⋅ a on both sides of the equation, we get the desired result. The proof that a ⋅ 0 = 0 is similar.
Corollary 1: A ring, (R, +, ⋅) is trivial (that is, consists of precisely one element) if and only if 0 = 1.
Suppose 1 = 0. Let a be in R; then a ⋅ 1 = 1 ⋅ a = a. Since 1 = 0, a ⋅ 0 = 0 ⋅ a = a. But a ⋅ 0 = 0 ⋅ a = 0. Therefore, a = 0. Since a was arbitrary, R consists of only one element, namely 0. Therefore, (R, +, ⋅) is the trivial ring. Conversly, if (R, +, ⋅) is trivial, it must contain precisely one element. Therefore, 0 = 1 (otherwise R would consist of at least two elements).
Theorem 2: (−1)a = −a
(−1)a + a = (−1)a + (1)a = (-1 + 1)a = (0)a = 0
. Similarly, a + (−1)a = 0, so that (−1)a is the additive inverse of a. Since the additive inverse of a is necessarily unique, (−1)a = −a, as desired.
Theorem 3: (−a) ⋅ b = a ⋅ (−b) = −(ab)
(−a) ⋅ b = (-1) ⋅ (a ⋅ b) = a⋅ (-1) ⋅ b = a(−b)
Similarly,
a(−b) = a ⋅ (-1) ⋅ b = (-1) ⋅ a ⋅ b = −(ab)
as desired.
Let (R, +, · ) be a ring. Then (R, +, · ) is said to be a zero ring if the product of any two elements in R is 0 (the additive identity).
Note
Example
An element a of a ring (R, +, · ) need not necessarily have a multiplicative inverse (for example, in the ring of all integers, 2 has no multiplicative inverse). An element a in a ring is called a unit if it is invertible with respect to multiplication. Formally,
Formal definition
Let (R, +, · ) be a ring. An element a of (R, +, · ) is said to be a unit in (R, +, · ) if there is an element b in the ring such that a · b = b · a = 1.
Notes
Examples
A homomorphism of rings is a function f: R1 → R2 between two rings R1 and R2 preserving the ring operations, in the sense that for all a, b in R1 the following identities are required to hold:
As in any category a map f possessing an inverse g: R2 → R1, i.e. a map in the opposite direction such that the two compositions f ○ g and g ○ f equal the identity map of R2 and R1, respectively, is called an isomorphism. Equivalently, f is bijective.
For any ring R with unit element, there is a natural map Z → R, it maps any positive n to the n-fold sum of the unit element of R, and −n to the additive inverse of n · 1R. The characteristic of R can be expressed in terms of this map.
Informally, a subring is a ring, (S, +, · ), contained in a bigger one, (R, +, · ).[8] More formally, let (R, +, · ) be a ring. A subset S of R is said to be a subring of R if:
If S is a subring of R, then S is a ring in its own right with + and · restricted to the cartesian product S X S.
The purpose of an ideal in a ring is to somehow allow one to define the quotient ring of a ring (analogous to the quotient group of a group; see below). An ideal in a ring can therefore be thought of as a generalization of a normal subgroup in a group. More formally, let (R, +, · ) be a ring. A subset I of R is said to be a right ideal in R if:
A left ideal is similarly defined with the second condition being replaced. More specifically, a subset I of R is a left ideal in R if:
Informally, the quotient ring of a ring, is a generalization of the notion of a quotient group of a group. More formally, given a ring (R, +, · ) and a two-sided ideal I of (R, +, · ), the quotient ring (or factor ring) R/I is the set of cosets of I (with respect to thee underlying adSource: this wikipedia article, under GFDL.