ring

An image illustrating geometric addition on a cubic curve in projective space. Ring theory has many important applications in algebraic geometry.

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, apart from the ring axioms themselves, rings have very little in common with the integers. 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.

Definition and illustration

First example: the integers

The most familiar example of a ring is the set of all integers, Z, consisting of the numbers

..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... ^[1]

together with the usual operations of addition and multiplication. These operations satisfy the following properties:

• The integers form an abelian group under addition; that is:
  • Closure axiom for addition: Given two integers a and b, their sum, a + b is also an integer.
  • Existence of additive identity: For any integer a, a + 0 = 0 + a = a. Zero is called the identity element of the integers because adding 0 to any integer (in any order) returns the same integer.
  • Commutativity of addition: For any two integers a and b, a + b = b + a. So the order in which you add two integers is irrelevant.
  • Associativity of addition: For any integers, a, b and c, (a + b) + c = a + (b + c). So, adding b to a, and then adding c to this result, is the same as adding c to b, and then adding this result to a.
  • Existence of additive inverse: For any integer a, there exists an integer denoted by −a such that a + (−a) = (−a) + a = 0. The element, −a, is called the additive inverse of a because adding a to −a (in any order) returns the identity.

• The integers form a multiplicative monoid (a monoid under multiplication); that is:
  • Closure axiom for multiplication: Given two integers a and b, their product, a · b is also an integer.
  • Associativity of multiplication: Given any integers, a, b and c, (a · b) · c = a · (b · c). So multiplying b with a, and then multiplying c to this result, is the same as multiplying c with b, and then multiplying a to this result.
  • Existence of multiplicative identity: For any integer a, a · 1 = 1 · a = a. So multiplying any integer with 1 (in any order) gives back that integer. One is therefore called the multiplicative identity.
• Multiplication is distributive over addition : These two structures on the integers (addition and multiplication) are compatible in the sense that
  • a · (b + c) = (a · b) + (a · c), and
  • (a + b) · c = (a · c) + (b · c)

for any three integers a, b and c.

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. To qualify as a ring, the set and two operations, (R, +, · ), must satisfy the following requirements known as the ring axioms. ^[2]

• (R, +, · ) is required to be an abelian group under addition:

1.	Closure under addition.	For all a, b in R, the result of the operation a + b is also in R.c[›]
2.	Associativity of addition.	For all a, b and c in R, the equation (a + b) + c = a + (b + c) holds.
3.	Existence of additive identity.	There exists an element 0 in R, such that for all elements a in R, the equation 0 + a = a + 0 = a holds.
4.	Existence of additive inverse.	For each a in R, there exists an element b in R such that a + b = b + a = 0
5.	Commutativity of addition.	For all a, b in R, the equation a + b = b + a holds.

• (R, +, · ) is required to be a monoid under multiplication:

1.	Closure under multiplication.	For all a, b in R, the result of the operation a · b is also in R.c[›]
2.	Associativity of multiplication.	For all a, b, and c in R, the equation (a · b) · c = a · (b · c) holds.
3.	Existence of multiplicative identity.	There exists an element 1 in R, such that for all elements a in R, the equation 1 · a = a · 1 = a holds.

• The distributive laws:

1. For all a, b and c in R, the equation a · (b + c) = (a · b) + (a · c) holds.

2. For all a, b and c in R, the equation (a + b) · c = (a · c) + (b · c) holds.

Notes on the definition

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.

Second example: the ring Z₄

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):

• For any x, y in Z₄, x + y is defined to be their sum in Z (the set of all integers) mod 4. So we can represent the additive structure of Z₄ by the left-most table as shown.

• For any x, y in Z₄, x ⋅ y is defined to be their product in Z (the set of all integers) mod 4. So we can represent the multiplicative structure of Z₄ by the right-most table as shown.

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

• In general, given any two integers, x and y, if x ⋅ y = 0, then either x is 0 or y is 0. It is interesting to note that this does not hold for the ring (Z₄, +, ⋅):

2 ⋅ 2 = 0

although neither factor is 0. In general, a non-zero element a of a ring, (R, +, ⋅) is said to be a zero divisor in (R, +, ⋅), if there exists a non-zero element b of R such that a ⋅ b = 0. So in this ring, the only zero divisor is 2 (note that 0 ⋅ a = 0 for any a in a ring (R, +, ⋅) so 0 is not considered to be a zero divisor).

• A commutative ring which has no zero divisors is called an integral domain (see below). So Z, the ring of all integers (see above), is an integral domain (and therefore a ring), although Z₄ (the above example) does not form an integral domain (but is still a ring). So in general, every integral domain is a ring but not every ring is an integral domain.

Third example: the trivial 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).

History

A portrait of Richard Dedekind: the founder of ring theory.

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]

Simple consequences of the ring axioms

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.

Basic properties of rings

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

Corollary 1: A ring, (R, +, ⋅) is trivial (that is, consists of precisely one element) if and only if 0 = 1.

Theorem 2: (−1)a = −a

Theorem 3: (−a) ⋅ b = a ⋅ (−b) = −(ab)

Group properties of rings

• An integer in a ring is an element n of the ring that can be written either as:

n = 1 + ... + 1 or,

n = −1 − ... − 1

Note that for any element a of a ring (R, +, ⋅), a ⋅ n = n ⋅ a for any integer n. In words, this means that the order in which you multiply an integer with another element in a ring is irrelevant.

• If a ring is a cyclic group under addition, then it is commutative (see above). More generally, the center of a ring contains the set of all integers of the ring. In particular, no ring can have trivial center unless 1 + 1 = 1, or, equivalently, 1 = 0 and the ring is trivial.

Binomial theorem for rings

• The binomial theorem

holds whenever x and y commute (see this for a proof). The theorem holds for arbitrary x and y in a commutative ring.

Basic concepts

Zero ring

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

• If we assume that rings have multiplicative identities, then a ring is zero if and only if it is trivial. Therefore, the concept of a zero ring is only interesting when one deals with rings that are only semigroups under multiplication (that is, do not necessarily have a multiplicative identity).

Example

• Any abelian group can be turned into a zero ring by letting a · b = 0 for every a, b in the group (and letting + be the group operation of that abelian group).

Units

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

• Using elementary group theory, one can deduce that b is uniquely determined by a so let 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*.

Examples

• Every non-zero element in the ring of all real numbers ((R, +, · )) is a unit in this ring. In fact, one can define a field (see below) to be a commutative ring such that every non-zero element is a unit. Therefore, (R, +, · ) forms a field.

Opposite ring

• Let (R, +, · ) be a ring. The opposite ring, (R^op, +, * ) is defined by a * b = b · a. Addition on the opposite ring is defined to be the same as addition on the original ring.

Note

• If a ring (R, +, · ) is commutative, then it is isomorphic to its opposite ring. The isomorphism is given by the identity map (in the category of sets).

Examples

• If the opposite ring of a ring (R, +, · ) is commutative, then so is the original ring.

• Consider the opposite ring of the opposite ring; that is R^opop. Note that, R^opop = R so that the operation on rings, taking any ring to its 'opposite' is involutory.

• Note that, in particular, the above example shows that one can define a natural functor from the category of rings to itself.

• If two rings R₁ and R₂ are isomorphic, then their corresponding opposite rings are also isomorphic.

Ring homomorphisms

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:

• f(a + b) = f(a) + f(b)

• f(a · b) = f(a) · f(b)

• f (1_R1) = 1_R2

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 · 1_R. The characteristic of R can be expressed in terms of this map.

Subrings

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:

• For every a, b in S, a + b is in S

• For every a, b in S, a · b is in S

• For every a in S, −a (the additive inverse of a in R) is in S

• The multiplicative identity of R is in S, i.e 1 is in S

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.

Examples

• Suppose is a ring morphism. Then the image, f (R₁) is a subring of R₂.

• If A is a subring of R and B is a subring of R such that B is a subset of A, then B is a subring of A.

Ideals

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:

• (I, +) is a subgroup of the underlying additive group in (R, +, · ) (i.e (I, +) is a subgroup of (R, +)).

• For every x in I and r in R, x · r is in I.

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:

• (I, +) is a subgroup of the underlying additive group in (R, +, · ) (i.e (I, +) is a subgroup of (R, +)).

• For every x in I and r in R, r · x is in I.

Notes

• If k is in R, then k · R is a right ideal in R, and R · k is a left ideal in R. These ideals (for any k in R) are called the principal right and left ideals generated by k.

• If every ideal in a ring (R, +, · ) is a principal ideal in (R, +, · ), (R, +, · ) is said to be a principal ideal ring.

• An ideal in a ring, (R, +, · ), is said to be a two-sided ideal if it is both a left ideal and right ideal in (R, +, · ). It is preferred to call a two-sided ideal, simply an ideal.

• If I = {0} (where 0 is the additive identity of the ring (R, +, · )), then I is an ideal known as the trivial ideal. Similarly, R is also an ideal in (R, +, · ) called the unit ideal.

Examples

• Any additive subgroup of the integers is an ideal in the integers with its natural ring structure.

• There are no non-trivial ideals in R (the ring of all real numbers) (i.e, the only ideals in R are {0} and R itself). More generally, a field cannot contain any non-trivial ideals.

• From the previous example, every field must be a principal ideal ring.

• A subset, I, of a commutative ring (R, +, · ) is a left ideal if and only if it is a right ideal. So for simplicity's sake, we refer to any ideal in a commutative ring as just an ideal.

Quotient ring

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 the underlying additive group of (R, +, · ); i.e cosets with respect to (R, +)) together with the operations:

(a + I) + (b + I) = (a + b) + I and

(a + I)(b + I) = (ab) + I.

For every a, b in R.

Direct product of rings

Let R and S be rings. Then the product R X S can be equipped with the following natural ring structure:

• (r₁, s₁) + (r₂, s₂) = (r₁ + r₂, s₁ + s₂)

• (r₁, s₁) · (r₂, s₂) = (r₁ · r₂, s₁ · s₂)

for every r₁, r₂ in R and s₁, s₂ in S. The ring R X S with the above operations of addition and operation (derived from R and S) is called the direct product of R with S. By repeating this process n times with n rings, one can form what is known as the n-fold product of rings. The product is natural because the quotient ring R X S / R is isomorphic to S and similarly R X S / S is isomorphic to R.

Center of a ring

There are various notions to address the non-commutativity of general rings R. The center consists of the ring elements that commute with every other element:

C = {c in R | c · x = x · c for every x in R}

It is a subring of R.^{[citation needed]} A subring of R is said to be central in R, if it is a subset of C (and therefore a subring of C).

Boolean rings

A ring R is called Boolean ring if every ring element a is an idempotent, i.e. a · a = a. For example, the (unique) ring with two elements has this property. There is a one-to-one correspondence between Boolean algebras and Boolean rings, by expressing set difference and set intersection in terms of addition and multiplication and vice versa. Any Boolean ring is commutative and of characteristic two, i.e. a + a = 0 for all a in R.

Venn diagrams for the Boolean operations of conjunction, disjunction, and complement

Endomorphism rings

Let (A, +) be an abelian group and let E be the set of all endomorphisms of A (that is, the set of all group morphisms from A to itself) . If f and g belong to E, then define f + g to be the point-wise sum of f and g (with respect to the group operation). Similarly, define f · g = f o g (f composed with g). Then (E, +, · ) forms a ring known as the endomorphism ring of (A, +).

Example

• Note that one can define a natural functor from the category of abelian groups (Ab) to the category of rings using the notion of endomorphism rings.

• The endomorphism ring of an abelian group is trivial if and only if the abelian group in question is the trivial group.

More generally the endomorphism ring of any module is an important ring when studying the module. Every ring arises as the endomorphism ring of a module, the regular module. Conversely an R-module M is nothing more than a ring homomorphism from the ring R into the endomorphism ring of the underlying additive group of M. Properties of modules such as indecomposable and strongly indecomposable are often defined in terms of the module's associated endomorphism ring.

Tensor product of rings

• 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.

Examples and applications

Numbers

Many number systems such as the integers, rational numbers, real numbers and the complex numbers naturally occur as rings. Note that in the case of the rational, real and complex numbers, every non-zero element has a multiplicative inverse. These number systems (except for the integers) form what is known as a field (see below or at the top of the article for a comparison between fields and rings).

Rationals

The desire for the existence of multiplicative inverses in the ring of all integers (see above) suggests considering fractions

Fractions of integers (with b nonzero) are known as rational numbers.^d[›] The set of all such fractions is commonly denoted by Q. Note that the set of all rationals form a ring under the natural operations of addition and multiplication. In fact, the rationals satisfy a stronger axiom than the integers now that we have included fractions: namely, every non-zero rational has a multiplicative inverse, its reciprocal. Because of this, along with the fact that multiplication on the rationals is commutative (see above), the rationals form what is known as a field (see below) ^[9]. Fields are central objects to abstract algebra.

An important difference between the additive and multiplicative properties of the ring of integers is that multiplicative inverses are not required to exist. For instance, there is no integer a such that a · 2 = 1. This property distinguishes the integers from the rational numbers, which is also a set endowed with operations of addition and multiplication, satisfying all of the above properties but also the existence of a multiplicative inverse for each nonzero element: namely if a and b are nonzero integers the inverse of the rational number a/b is simply b/a. Although in some naive sense the rational numbers are "more complicated" than the integers (e.g. in the sense that a rational number can be viewed as an ordered pair of integers, subject to certain equivalences), in many respects the existence of multiplicative inverses gives the rational numbers a simpler structure. For instance, given a pair of integers a and b we may consider the relation that b is exactly divisible by a, i.e., there exists an integer c such that a · c = b, and using this we get a rich theory of prime and composite numbers, factorization, and so forth. In contrast any nonzero rational number divides any other nonzero rational number, a much less interesting relation. This is prototypical of the different and richer structure that an arbitrary ring has in comparison to a field or a division ring (see below).

Integers modulo a prime

Written Z / Zp, this ring is an elementary example of a finite field.

Polynomial rings

• The polynomial ring R[X] of polynomials over a ring R is also a ring.
  • The set of formal power series R[[X₁, …, X_n]] over a commutative ring R is a ring.

Some examples of the ubiquity of rings

It is remarkable how many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring. For instance:

• To any topological space X one can associate its integral cohomology ring , a graded ring. As the name suggests, there are also homology groups of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the spheres and tori, for which the methods of point-set topology are not well-suited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a k-multilinear form and an l-multilinear form to get a (k + l)-mulilinear form. The importance of the ring structure in cohomology is hard to overstate: it provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.
• To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.
• To any group ring or Hopf algebra is associated its representation ring or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from character theory, which is more or less the Grothendieck group given a ring structure.
• To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.
• Every simplicial complex has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in algebraic combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.

Finite rings

Every finite field is an example of a finite ring, and the additive part of every finite ring is an example of a finite group, but the concept of finite rings in their own right is much more complicated.

In 1964 David Singmaster proposed the following problem in the American Mathematical Monthly: "What is the smallest non-trivial ring with 1 that is not a field?" One can find the solution by D.M Bloom in a three page proof (71:918-20) that there are eleven rings of order 4, three of which have a 1.

The occasion of non-commutativity in finite rings was described in 1968 in the same journal (75:512-14) by K. Eldrige in two theorems: If the order m of a finite ring with 1 has a cube-free factorization, then it is commutative. And if a non-commutative finite ring with 1 has the order of a prime cubed, then the ring is isomorphic to the upper trianglar 2 x 2 matrix ring over the Galois field of the prime. The science of rings of order the cube of a prime was further developed by R. Raghavendra in 1969 (Compositio Mathematica 21:195-229). In 1973 the Proceedings of the Japan Academy 49:795-9 published Robert Gilmer and Joe Mott’s paper "Associative rings of order p³". Next Flor and Wessenbauer (1975) made improvements on the cube-of-a-prime case. Definitive work on the isomorphism classes came with V.G. Antipkin and V.P Elizarov (1982) writing in the Siberian Mathematical Journal (23:457-64). They prove that for p > 2, the number of classes is 3p + 50.

There are earlier references in the topic of finite rings, such as Robert Ballieu (1947) "Anneaux finis" in Ann. Soc. Sci Bruxelles (61:222-227). Earlier work by Scorza (1935) is noted by Irving Kaplansky in his review (MR0022841) of Ballieu.

These facts are known about the number of finite rings of a given order (suppose p and q represent prime numbers):

• There are eleven finite rings of order p².
• There are twenty-two finite rings of order p²q.
• There are fifty-two finite rings of order eight.
• There are 3 p + 50 finite rings of order p³, p > 2.

The number of rings with n elements is listed under A027623 in the On-Line Encyclopedia of Integer Sequences.

Rings with additional structure

Associative algebras

An associative algebra is a ring that is also a vector space over a field K. For instance, the set of n by n matrices over the real field R has dimension n² as a real vector space, and the matrix multiplication corresponds to the ring multiplication. For a non-trivial but elementary example consider real matrices (2 x 2).

Lie rings

A Lie ring is defined to be a ring that is nonassociative and anticommutative under multiplication, that also satisfies the Jacobi identity. More specifically we can define a Lie ring L to be an abelian group under addition with an operation that has the following properties:

• Bilinearity:

for all x, y, z ∈ L.

• Jacobi identity:

for all x, y, z in L.

• For all x in L.

Lie rings need not be Lie groups under addition. Any Lie algebra is an example of a Lie ring. Any associative ring can be made into a Lie ring by defining a bracket operator [x,y] = xy − yx. Conversely to any Lie algebra there is a corresponding ring, called the universal enveloping algebra.

Lie rings are used in the study of finite p-groups through the Lazard correspondence. The lower central factors of a p-group are finite abelian p-groups, so modules over Z/pZ. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the commutator of two coset representatives. The Lie ring structure is enriched with another module homomorphism, then pth power map, making the associated Lie ring a so-called restricted Lie ring.

Lie rings are also useful in the definition of a p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic integers. The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and the reducing modulo p to get a Lie algebra over a finite field.

Topological ring

Let (X, T) is a topological space and (X, +, · ) be a ring. Then (X, T, +, · ) is said to be a topological ring, if its ring structure and topological structure are both compatible (i.e work together) over each other. That is, the addition map ( ) and the multiplication map ( ) have to be both continuous as maps between topological spaces where X x X inherits the product topology. So clearly, any topological ring is a topological group (under addition).

Examples

• The set of all real numbers, R, with its natural ring structure and the standard topology forms a topological ring.

• The direct product of two topological rings is also a topological ring.

Commutative rings

Although ring addition is commutative, so that for every a, b in R, a + b = b + a, ring multiplication is not required to be commutative; a · b need not equal b · a for all a, b in R. Rings that also satisfy commutativity for multiplication are called commutative rings.^e[›] Formally,

Formal definition

Let (R, +, · ) be a ring. Then (R, +, · ) is said to be a commutative ring if for every a, b in R, a · b = b · a. That is, (R, +, · ) is required to be a commutative monoid under multiplication.

Examples

• The integers form a commutative ring under the natural operations of addition and multiplication.

• An example of a non-commutative ring is the ring of n × n matrices over a non-trivial field K, for n > 1. In particular, the ring of all 2 × 2 matrices over R (the set of all real numbers) do not form a commutative ring as the following computation shows:

, which is not equal to

Principal ideal domains

Although rings are structurally similar to the integers, there are certain ring-theoretic properties that the integers may satisfy but a general ring may not. One such property is the requirement that every ideal in a ring be generated by a single element; that is, be a principal ideal. Formally,

Definition

Let R be a ring. Then R is said to be a principal ideal ring (abbreviated PIR), if every ideal in R is of the form a · R = {a · r | r · R}. A principal ideal domain is a principal ideal ring that is also an integral domain.

The requirement that a ring be a principal ideal domain is somewhat stronger than the other more common properties a ring may satisfy. For example, it is true that if a ring, R, is a unique factorization domain (UFD), then the polynomial ring over R is also a UFD. However, such a result does not in general hold for principal ideal rings. For example, the integers are an easy example of a principal ideal ring, but the polynomial ring over the integers fails to be a PIR; if R = Z[x] denotes the polynomial ring over the integers, I = 2 · R + X · R is an ideal which cannot be generated by a single element. Despite this counterexample, the polynomial ring over any field is always a principal ideal domain and in fact, a Euclidean domain. More generally, a polynomial ring is a PID if and only if the polynomial ring in question is over a field.

Aside from the polynomial ring over a PIR, principal ideal rings possess many interesting properties due to their connection with the integers in terms of divisibility; that is, principal ideal domains behave similarly to the integers with respect to divisibility. For example, any PIR is a UFD; i.e, an analogue of the fundamental theorem of arithmetic holds for principal ideal domains. Furthermore, since Noetherian rings are precisely those rings in which any ideal is finitely generated, principal ideal domains are trivially Noetherian rings. The fact that irreducible elements coincide with prime elements for PID's, together with the fact that every PID is Noetherian, implies that any PID is a UFD. One can also speak of the greatest common divisor of two elements in a PID; if x and y are elements of R, a principal ideal domain, then x · R + y · R = c · R for some c in R since the left-hand side is indeed an ideal. Therefore, c is the desired "GCD" of x and y.

An important class of rings lying in between fields and PID's, is the class of Euclidean domains. In particular, any field is a Euclidean domain, and any Euclidean domain is a PID. An ideal in a Euclidean domain is generated by any element of that ideal with minimum degree (all such elements must be associate). However, not every PID is a Euclidean domain; the ring furnishes a counterexample.

Unique factorization domains

The theory of unique factorization domains (UFD) also form an important part of ring theory. In effect, a unique factorization domain is ring in which an analogue of the fundamental theorem of arithmetic holds. Formally,

Definition

Let R be a ring. Then R is said to be a unique factorization domain (abbreviated UFD), if the following conditions are satisfied:

1. R is an integral domain.

2. Every non-zero non-unit of R is the product of a finite number of irreducible elements.

3. If = , where all a_i's and b_j's are irreducible, then n = m and after possible renumbering of the a_i's and b_j's, b_i = a_i · u_i where u_i is a unit in R.

The second condition above guarantees that "non-trivial" elements of R can be decomposed into irreducibles, and according to the third condition, such a finite decomposition is unique "up to multiplication by unit elements." This weakened form of uniqueness is reasonable to assume for otherwise even the integers would not satisfy the properties of being a UFD ((-2)² = 2² = 4 demonstrates two "distint" decompositions of 4; however both decompositions of 4 are equivalent up to multiplication by units (-1 and +1)). The fact that the integers constitute a UFD follows from the fundamental theorem of arithmetic.

For arbitrary rings, one may define a prime element and an irreducible element; these two may not in general coincide. However, a prime element in a domain is always irreducible. For UFD's, irreducible elements are also primes.

The class of unique factorization domains is related to other classes of rings. For instance, any Noetherian domain satisfies conditions 1 and 2 above, but in general Noetherian domains fail to satisfy condition 3. However, if the set of prime elements and the set of irreducible elements coincide for a Noetherian domain, the third condition of a UFD is satisfied. In particular, principal ideal domains are UFD's.

Integral domains and fields

While rings are very important mathematical objects, there are a lot of restrictions involved in their theory. For instance, suppose we have a ring R and suppose a and b are in R. If a is non-zero and a · b = 0, then b need not necessarily be 0. In particular, if a · b = a · c with a non-zero b need not equal c. An example of this is the set of n x n matrices over R where a maybe such a non-zero matrix but may be singular. In this case, the result may not be true. However, we can impose additional conditions on the ring to ensure that this be true; namely make the ring into an integral domain (that is a non-trivial commutative ring with no zero divisors). But we still run into a problem; namely we can't necessarily divide by non-zero elements. For example, the collection of all integers form an integral domain but we still can't divide an integer a by another integer b. For example, 2 cannot divide 3 to obtain another element in this ring. However, this problem can readily be solved if we ensure that every element in the ring has a multiplicative inverse. A field is a ring in which the non-zero elements form an abelian group under multiplication. In particular, a field is an integral domain (and therefore has no zero divisors) along with an additional operation of 'division'. Namely, if a and b are in a field F, then a/b is defined to be a · b^-1 which is well defined.

Formal definition

Let (R, +, · ) be a ring. Then (R, +, · ) is said to be an integral domain if (R, +, · ) is commutative and has no zero divisors. Furthermore, (R, +, · ) is said to be a field, if its non-zero elements form an abelian group under multiplication.

Note

• If we require that the zero element (0) in a ring to have a multiplicative inverse, then the ring must be trivial.

Examples

• The integers form a commutative ring under the natural operations of addition and multiplication. In fact, the integers form what is known as an integral domain (a commutative ring with no zero divisors).

• A ring whose non-zero elements form an abelian group under multiplication (not just a commutative monoid), is called a field. So every field is an integral domain and every integral domain is a commutative ring. Furthermore, any finite integral domain is a field.

Relation to other algebraic structures

The following is a chain of class inclusions that describes the relationship between rings, domains and fields:

• Commutative rings ⊃ integral domains ⊃ half factorization domains ⊃unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields

Fields and integral domains are very important in modern algebra.

Non-commutative rings

The study of non-commutative rings is a major area in modern algebra; especially ring theory. Often non-commutative rings possess interesting invariants that commutative rings do not. As an example, there exist rings which contain non-trivial proper left or right ideals, but are still simple; that is contain no non-trivial proper (two-sided) ideals. This example illustrates how one must take care when studying non-commutative rings because of possible counterintuitive misconceptions.

The theory of vector spaces is one illustration of a special case of an object studied in non-commutative ring theory. In linear algebra, the "scalars of a vector space" are required to lie in a field - a commutative division ring. The concept of a module, however, requires only that the scalars lie in an abstract ring. Neither commutativity nor the division ring assumption is required on the scalars in this case. Module theory has various applications in non-commutative ring theory, as one can often obtain information about the structure of a ring by making use of its modules. The concept of the Jacobson radical of a ring; that is, the interesection of all right/left annhiliators of simple right/left modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right/left ideals in the ring, shows how the internal structure of the ring is reflected by its modules. It is also remarkable that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; whether commutative or non-commutative. Therefore, the Jacobson radical also captures a concept which may seem to be not well-defined for non-commutative rings.

Non-commutative rings serve as an active area of research due to their ubiquity in mathematics. For instance, the ring of n by n matrices over a field is non-commutative despite its natural occurence in physics. More generally, endomorphism rings of abelian groups are rarely commutative, and in fact, the naturally occuring ring of all continuous functions from R (the set of all real numbers) into R, equipped with point-wise addition and function composition (as multiplication) is certainly not commutative.

Non-commutative rings, like non-commutative groups, are not very well understood. For instance, although every finite abelian group is the direct sum of (finite) cyclic groups of prime-power order, non-abelian groups do not possess such a simple structure. Likewise, various invariants exist for commutative rings, whereas invariants of non-commutative rings are difficult to find. As an example, the nilradical, although "innocent" in nature, need not be an ideal unless the ring is assumed to be commutative. Specifically, the set of all nilpotent elements in the ring of all n x n matrices over a division ring never forms an ideal, irrespective of the division ring chosen. Therefore, the nilradical cannot be studied in non-commutative ring theory. Note however that there are analogues of the nilradical defined for non-commutative rings, that coincide with the nilradical when commutativity is assumed.

Category theoretical description

Every ring can be thought of as monoids in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product). The monoid action of a ring R on a abelian group is simply an R-module. Essentially, an R-module is a generalization of the notion of a vector space - where rather than a vector space over a field, one has a "vector space over a ring".

Let (A, +) be an abelian group and let End(A) be its endomorphism ring (see above). Note that, essentially, End(A) is the set of all morphisms of A, where if f is in End(A), and g is in End(A), the following rules may be used to compute f + g and f · g:

• (f + g)(x) = f(x) + g(x)

• (f · g)(x) = f(g(x))

where + as in f(x) + g(x) is addition in A, and function composition is denoted from right to left. Therefore, associated to any abelian group, is a ring. Conversly, given any ring, (R, +, · ), (R, +) is an abelian group. Furthermore, for every r in R, right (or left) multiplication by r gives rise to a morphism of (R, +), by right (or left) distributivity. Let A = (R, +). Consider those endomorphisms of A, that "factor through" right (or left) multiplication of R. In other words, let End_R(A) be the set of all morphisms m of A, having the property that m(r · x) = r · m(x). It was seen that every r in R gives rise to a morphism of A - right multiplication by r. It is in fact true that this assocation of any element of R, to a morphism of A, as a function from R to End_R(A), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian X-group (by X-group, it is meant a group with X being its set of operators). In essence, the most general form of a ring, is the endomorphism group of some abelian X-group.

See also

Wikibooks has a book on the topic of Abstract algebra/Rings, fields and modules

Notes

Citations

References

General references

• R.B.J.T. Allenby (1991). Rings, Fields and Groups. Butterworth-Heinemann. ISBN 0-340-54440-6. 
• Atiyah M. F., Macdonald, I. G., Introduction to commutative algebra. Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. 1969 ix+128 pp.
• Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, 1999.
• 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. 
• Dresden, G. "Small Rings." [2]
• Ellis, G. Rings and Fields. Oxford, England: Oxford University Press, 1993.
• Goodearl, K. R., Warfield, R. B., Jr., An introduction to noncommutative Noetherian rings. London Mathematical Society Student Texts, 16. Cambridge University Press, Cambridge, 1989. xviii+303 pp. ISBN 0-521-36086-2
• Herstein, I. N., Noncommutative rings. Reprint of the 1968 original. With an afterword by Lance W. Small. Carus Mathematical Monographs, 15. Mathematical Association of America, Washington, DC, 1994. xii+202 pp. ISBN 0-88385-015-X
• Nagell, T. "Moduls, Rings, and Fields." §6 in Introduction to Number Theory. New York: Wiley, pp. 19-21, 1951
• Nathan Jacobson, Structure of rings. American Mathematical Society Colloquium Publications, Vol. 37. Revised edition American Mathematical Society, Providence, R.I. 1964 ix+299 pp.
• Nathan Jacobson, The Theory of Rings. American Mathematical Society Mathematical Surveys, vol. I. American Mathematical Society, New York, 1943. vi+150 pp.
• Kaplansky, Irving (1974), Commutative rings (Revised ed.), University of Chicago Press, MR0345945 
• Lam, T. Y., A first course in noncommutative rings. Second edition. Graduate Texts in Mathematics, 131. Springer-Verlag, New York, 2001. xx+385 pp. ISBN 0-387-95183-0
• Lam, T. Y., Exercises in classical ring theory. Second edition. Problem Books in Mathematics. Springer-Verlag, New York, 2003. xx+359 pp. ISBN 0-387-00500-5
• Lam, T. Y., Lectures on modules and rings. Graduate Texts in Mathematics, 189. Springer-Verlag, New York, 1999. xxiv+557 pp. ISBN 0-387-98428-3
• Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, 211 (Revised third ed.), New York: Springer-Verlag, MR1878556, ISBN 978-0-387-95385-4 

• Lang, Serge (2005), Undergraduate Algebra (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-22025-3 .
• Matsumura, Hideyuki (1989), Commutative Ring Theory, Cambridge Studies in Advanced Mathematics (2nd ed.), Cambridge University Press, ISBN 978-0-521-36764-6 
• McConnell, J. C.; Robson, J. C. Noncommutative Noetherian rings. Revised edition. Graduate Studies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001. xx+636 pp. ISBN 0-8218-2169-5
• Pinter-Lucke, James (2007), "Commutativity conditions for rings: 1950–2005", Expositiones Mathematicae 25 (2): 165–174, doi:10.1016/j.exmath.2006.07.001, ISSN 0723-0869 
• Rowen, Louis H., Ring theory. Vol. I, II. Pure and Applied Mathematics, 127, 128. Academic Press, Inc., Boston, MA, 1988. ISBN 0-12-599841-4, ISBN 0-12-599842-2
• Sloane, N. J. A. Sequences A027623 and A037234 in "The On-Line Encyclopedia of Integer Sequences
• Zwillinger, D. (Ed.). "Rings." §2.6.3 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 141-143, 1995

Special references

• Balcerzyk, Stanisław; Józefiak, Tadeusz (1989), Commutative Noetherian and Krull rings, Ellis Horwood Series: Mathematics and its Applications, Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155615-7 

• Balcerzyk, Stanisław; Józefiak, Tadeusz (1989), Dimension, multiplicity and homological methods, Ellis Horwood Series: Mathematics and its Applications., Chichester: Ellis Horwood Ltd., ISBN 978-0-13-155623-2 
• Ballieu, R. "Anneaux finis; systèmes hypercomplexes de rang trois sur un corps commutatif." Ann. Soc. Sci. Bruxelles. Sér. I 61, 222-227, 1947.
• Berrick, A. J. and Keating, M. E. An Introduction to Rings and Modules with K-Theory in View. Cambridge, England: Cambridge University Press, 2000.
• Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry., Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, MR1322960, ISBN 978-0-387-94268-1; 978-0-387-94269-8 
• Fine, B. "Classification of Finite Rings of Order ." Math. Mag. 66, 248-252, 1993
• Fletcher, C. R. "Rings of Small Order." Math. Gaz. 64, 9-22, 1980
• Fraenkel, A. "Über die Teiler der Null und die Zerlegung von Ringen." J. reine angew. Math. 145, 139-176, 1914
• Gilmer, R. and Mott, J. "Associative Rings of Order ." Proc. Japan Acad. 49, 795-799, 1973
• Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, 1998
• Jacobson, Nathan (1945), "Structure theory of algebraic algebras of bounded degree", Annals of Mathematics 46 (4): 695–707, ISSN 0003-486X 
• Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998
• Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New York: Dover, 2000
• Nagata, Masayoshi (1962), Local rings, Interscience Tracts in Pure and Applied Mathematics, 13, Interscience Publishers, pp. xiii+234, MR0155856, ISBN 978-0-88275-228-0 (1975 reprint) 
• Pierce, Richard S., Associative algebras. Graduate Texts in Mathematics, 88. Studies in the History of Modern Science, 9. Springer-Verlag, New York-Berlin, 1982. xii+436 pp. ISBN 0-387-90693-2
• Zariski, Oscar; Samuel, Pierre (1975), Commutative algebra, Graduate Texts in Mathematics, 28, 29, Berlin, New York: Springer-Verlag 

Historical references

• History of ring theory at the MacTutor Archive
• Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillian, 1996
• Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 4th ed. New York: Springer-Verlag, 2004. ISBN 3-540-43491-7
• Faith, Carl, Rings and things and a fine array of twentieth century associative algebra. Mathematical Surveys and Monographs, 65. American Mathematical Society, Providence, RI, 1999. xxxiv+422 pp. ISBN 0-8218-0993-8
• Itô, K. (Ed.). "Rings." §368 in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, 1986
• Kleiner, I. "The Genesis of the Abstract Ring Concept." Amer. Math. Monthly 103, 417-424, 1996
• Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24-34, 2005
• Singmaster, D. and Bloom, D. M. "Problem E1648." Amer. Math. Monthly 71, 918-920, 1964
• Van der Waerden, B. L. A History of Algebra. New York: Springer-Verlag, 1985
• Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002