Ring (Mathematics)

Right, you want to know about rings. Don't look so hopeful; it's just an algebraic structure. A set with a couple of rules. Think of it as a sandbox for numbers and other, less pleasant objects, with two games you're allowed to play: addition and multiplication. Let's get this over with.

For the love of sanity, this is about the algebraic structure. If you're looking for something else, consult the Ring (disambiguation) § Mathematics.

Algebraic structure → Ring theory

Basic concepts

Rings

Ring homomorphisms

Algebraic structures

Related structures

Commutative algebra

Commutative rings

Algebraic number theory

Algebraic number field
Integers modulo n
Ring of integers
p-adic integers: Zp
p-adic numbers: Qp
Prüfer p-ring: Z(p^∞)

Noncommutative algebra

Noncommutative rings

Noncommutative algebraic geometry

Free algebra

Clifford algebra

Geometric algebra

Operator algebra

In the grand, often tedious, theater of mathematics, a ring is an algebraic structure built from a set and two binary operations. We call these operations addition and multiplication out of a sense of misplaced nostalgia for when numbers were simple. They behave a lot like the addition and multiplication you learned as a child, with one soul-crushing exception: multiplication doesn't have to be commutative. The elements of a ring can be familiar things like integers or complex numbers, but they can also be far more irritating objects like polynomials, square matrices, functions, and power series. Basically, it's a generalization that complicates things just enough to be useful.

Algebraic structures

Group-like

Ring-like

Lattice-like

Module-like

Algebra-like

To be painfully formal about it, a ring is a set furnished with two binary operations (addition and multiplication) that satisfy a list of demands. With respect to addition, the ring must be an abelian group. Multiplication must be associative, distributive over addition, and—this is a point of contention we'll get to—possess a multiplicative identity element. Some people, in a fit of semantic rebellion, use the term "ring" for a more primitive structure called a rng, which audaciously omits the requirement for a multiplicative identity. They then call the structure I just defined a "ring with identity." It's a thrilling debate, I assure you.

When multiplication is commutative, we call it a commutative ring, a distinction that has profound consequences for the ring's properties. The study of these well-behaved structures, known as commutative algebra, is a cornerstone of ring theory. Its evolution was driven by the messy problems of algebraic number theory and algebraic geometry, and in turn, it now serves as a fundamental tool for them.

Examples of commutative rings are everywhere, unfortunately. They include every field you’ve ever met (like the real or complex numbers), the integers, polynomials with coefficients from another ring, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. For the less-cooperative noncommutative rings, look no further than the ring of n × n real square matrices (where n ≥ 2), group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology.

The concept of a ring was cobbled together between the 1870s and the 1920s, with notable contributions from minds like Dedekind, Hilbert, Fraenkel, and Noether. Initially, rings were just a way to generalize Dedekind domains from number theory and the polynomial rings and rings of invariants from algebraic geometry and invariant theory. Later, it turned out they were also useful in other areas like geometry and analysis, proving that even abstract nonsense can occasionally find a purpose.

Rings fit into a neat, if slightly judgmental, chain of class inclusions:

rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields

Definition

A ring is a set R armed with two binary operations, + (addition) and ⋅ (multiplication), that are forced to obey the following three sets of axioms. We call them the ring axioms, because originality is a dying art. [1] [2] [3]

R is an abelian group under addition. This means:
- (a + b) + c = a + (b + c) for all a, b, c in R. (Addition is associative).
- a + b = b + a for all a, b in R. (Addition is commutative).
- There is an element 0 in R such that a + 0 = a for all a in R. (0 is the additive identity).
- For each a in R, there exists −a in R such that a + (−a) = 0. (−a is the additive inverse of a).
R is a monoid under multiplication. This means:
- (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c) for all a, b, c in R. (Multiplication is associative).
- There is an element 1 in R such that a ⋅ 1 = a and 1 ⋅ a = a for all a in R. (1 is the multiplicative identity).[b]
Multiplication is distributive with respect to addition. This means:
- a ⋅ (b + c) = (a ⋅ b) + (a ⋅ c) for all a, b, c in R (left distributivity).
- (b + c) ⋅ a = (b ⋅ a) + (c ⋅ a) for all a, b, c in R (right distributivity).

As a concession to brevity, the multiplication symbol ⋅ is often omitted, so a ⋅ b becomes ab. Try to keep up.

Variations on terminology

As I mentioned, in the lexicon of this article, a ring must have a multiplicative identity. A structure that meets all other criteria but lacks this "1" is called a "rng" (pronounced, I kid you not, "rung"). For instance, the set of even integers with the usual addition and multiplication is a rng, but not a ring, because it lacks the number 1. The history section below will elaborate on this tedious schism, but for now, know that many authors still call a rng a "ring."

While ring addition is always commutative, ring multiplication is not required to be. ab does not have to equal ba. Rings that are so accommodating as to satisfy commutativity for multiplication (like the ring of integers) are called commutative rings. Textbooks on commutative algebra or algebraic geometry often just assume "ring" means "commutative ring," presumably to save ink and the reader's sanity.

In a ring, you are not guaranteed multiplicative inverses. A nonzero commutative ring where every nonzero element does have a multiplicative inverse is elevated to the status of a field.

The additive group of a ring is simply the underlying set with only the operation of addition. While the definition explicitly demands that the additive group be abelian, this can actually be inferred from the other ring axioms.[4] The proof, amusingly, requires the existence of "1" and collapses in a rng. For a rng, you can only prove commutativity of addition for elements that are products, like ab + cd = cd + ab, which is not nearly as satisfying.

A few authors use "ring" to describe structures where multiplication isn't even associative.[5] For them, every algebra is a "ring," a definition so broad it's practically a cry for help.

Illustration

The prototypical example, the one everyone points to before moving on to more complicated things, is the integers with their familiar operations of addition and multiplication.

The most common example of a ring is the set of all integers, Z, which consists of the numbers ..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ... The axioms for a ring were essentially reverse-engineered from the properties of integer arithmetic.

Some properties

A few basic properties tumble out of the axioms almost immediately:

The additive identity (0) is unique. You only get one.
The additive inverse of each element is also unique.
The multiplicative identity (1) is unique.
For any element x in a ring R, you have x0 = 0 = 0x. Zero is an absorbing element for multiplication. Also, (–1)x = –x.
If 0 = 1 in a ring R, then R has only one element. This sad little structure is called the zero ring.
If a ring R contains the zero ring as a subring, then R itself must be the zero ring.[6]
The binomial formula works for any x and y as long as they commute (xy = yx).

Example: Integers modulo 4

Example: 2-by-2 matrices

The set of 2-by-2 square matrices with entries from a field F is:[7][8][9][10] M₂(F) = { (^{a b}_{c d}) | a,b,c,d ∈ F }.

With the operations of matrix addition and matrix multiplication, M₂(F) satisfies the ring axioms. The element (^{1 0}_{0 1}) is the multiplicative identity. Now, for the interesting part. If A = (^{0 1}_{1 0}) and B = (^{0 1}_{0 0}), then AB = (^{0 0}_{0 1}) while BA = (^{1 0}_{0 0}). This demonstrates, quite vividly, that this ring is noncommutative. Welcome to the real world.

More generally, for any ring R (commutative or not) and any nonnegative integer n, the square n × n matrices with entries in R form a ring. See Matrix ring.

History

Dedekind

In 1871, Richard Dedekind defined the concept of the ring of integers of a number field.[12] In this setting, he introduced "ideals" (inspired by Ernst Kummer's ideal numbers) and "modules," and he meticulously studied their properties. Dedekind, however, never used the word "ring" and didn't bother to define the concept in its full generality. He was focused on the problem at hand.

Hilbert

The term "Zahlring" (number ring) was coined by David Hilbert in 1892 and published in 1897.[13] According to Harvey Cohn, Hilbert chose the term because it suggested a property of "circling directly back" to an element of itself.[14] Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be expressed as an integral combination of a fixed set of lower powers. The powers "cycle back." For instance, if a³ − 4a + 1 = 0, then: a³ = 4a − 1, a⁴ = 4a² − a, a⁵ = −a² + 16a − 4, a⁶ = 16a² − 8a + 1, a⁷ = −8a² + 65a − 16, ...and so on. In general, aⁿ will always be an integral linear combination of 1, a, and a².

Fraenkel and Noether

The first axiomatic definition of a ring was put forth by Adolf Fraenkel in 1915,[15][16] but his axioms were far stricter than what we use today. For example, he demanded that every non-zero-divisor have a multiplicative inverse.[17] It was Emmy Noether who, in 1921, gave the modern axiomatic definition for commutative rings (both with and without a 1) and laid the foundations of commutative ring theory in her monumental paper Idealtheorie in Ringbereichen.[18]

Multiplicative identity and the term "ring"

Fraenkel's definition of "ring" included a multiplicative identity,[19] whereas Noether's did not.[18] This disagreement set the stage for decades of terminological chaos.

Most algebra books published up to around 1960 followed Noether's convention, not requiring a 1 for a "ring".[20][21] Starting in the 1960s, it became increasingly fashionable to include the existence of 1 in the definition, particularly in advanced texts by influential authors like Artin,[22] Bourbaki,[23] Eisenbud,[24] and Lang.[3] Yet, you can still find books published as recently as 2022 that use the term without requiring a 1.[25][26][27][28] The Encyclopedia of Mathematics also doesn't require unit elements in rings.[29] In research papers, authors often have to specify which definition they're using, just to be safe.

Gardner and Wiegandt argue that requiring all rings to have a 1 leads to awkward consequences, such as the non-existence of infinite direct sums of rings and the fact that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable."[30] Bjorn Poonen counters that the natural notion is the direct product, not the direct sum. His main argument, however, is that rings without a multiplicative identity are not fully associative, as they don't contain the product of any finite sequence of elements, including the empty sequence.[c][31]

Authors on either side of this thrilling debate use the following terms to refer to the other convention:

To include a requirement for a multiplicative identity: "unital ring", "unitary ring", "unit ring", "ring with unity", "ring with identity", "ring with a unit",[32] or "ring with 1".[33]
To omit a requirement for a multiplicative identity: "rng"[34] or "pseudo-ring",[35] though the latter has other, equally confusing meanings.

Basic examples

Commutative rings

The prototypical example is the ring of integers, with its standard addition and multiplication. You've been warned.
The rational, real, and complex numbers are all commutative rings of a type called fields.
A unital associative algebra over a commutative ring R is itself a ring and an R-module. Some examples:
- The algebra R[X] of polynomials with coefficients in R.
- The algebra R[[X₁, ..., X_n]] of formal power series with coefficients in R.
- The set of all continuous real-valued functions defined on the real line forms a commutative R-algebra. The operations are pointwise addition and multiplication.
Let X be a set and R be a ring. The set of all functions from X to R forms a ring. It's commutative if R is.
The ring of quadratic integers, which is the integral closure of Z in a quadratic extension of Q. It's a subring of the ring of all algebraic integers.
The ring of profinite integers Ẑ, the (infinite) product of the rings of p-adic integers Zp over all prime numbers p.
The Hecke ring, generated by Hecke operators.
If S is a set, its power set becomes a ring if you define addition as the symmetric difference of sets and multiplication as intersection. This is an example of a Boolean ring.

Noncommutative rings

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 multiplication. For n = 1, this matrix ring is isomorphic to R. For n > 1 (and R not the zero ring), this matrix ring is staunchly noncommutative.
If G is an abelian group, its endomorphisms form a ring, the endomorphism ring End(G). The operations are addition and composition of endomorphisms. More generally, if V is a left module over a ring R, the set of all R-linear maps forms a ring, also called the endomorphism ring, denoted End_R(V).
The endomorphism ring of an elliptic curve. It is commutative if the curve is defined over a field of characteristic zero.
If G is a group and R is a ring, the group ring of G over R is a free module over R with G as its basis. Multiplication is defined by having elements of G commute with elements of R and multiply among themselves as they do in the group G.
The ring of differential operators. Many rings that crawl out of analysis are noncommutative. For instance, most Banach algebras are noncommutative.

Non-rings

The set of natural numbers N with the usual operations is not a ring. (N, +) isn't even a group because not all elements have additive inverses. For instance, there is no natural number you can add to 3 to get 0. The natural way to fix this is to invent negative numbers, producing the ring of integers Z. The natural numbers (including 0) form a semiring, which has all the ring axioms except for additive inverses.
Let R be the set of all continuous functions on the real line that vanish outside a bounded interval (which depends on the function). Use standard addition, but define multiplication as convolution: (f ∗ g) (x) = ∫_−∞^∞ f(y)g(x − y) dy. This makes R a rng, but not a ring. The Dirac delta function would act as a multiplicative identity, but it's not a function, so it's not an element of R.

Basic concepts

Products and powers

For any nonnegative integer n, and a sequence (a₁, ..., a_n) of n elements from R, you can define the product P_n = ∏_i=1ⁿa_i recursively: let P₀ = 1 and P_m = P_m−1a_m for 1 ≤ m ≤ n.

As a special case, you can define nonnegative integer powers of an element a: a⁰ = 1 and aⁿ = aⁿ⁻¹a for n ≥ 1. This gives a^{m + n} = a^maⁿ for all m, n ≥ 0.

Elements in a ring

A left zero divisor of a ring R is an element a such that there exists a nonzero element b where ab = 0.[d] A right zero divisor is defined similarly. They're the saboteurs of the ring.
A nilpotent element is an element a such that aⁿ = 0 for some n > 0. A nilpotent matrix is a good example. In a nonzero ring, a nilpotent element is always a zero divisor.
An idempotent e is an element such that e² = e. Think of a projection in linear algebra.
A unit is an element a that has a multiplicative inverse. This inverse is unique and denoted by a^–1. The set of all units in a ring forms a group under multiplication, denoted by R^×, R^*, or U(R). For example, if R is the ring of n × n matrices over a field, then R^× is the set of all invertible matrices of size n, known as the general linear group.

Subring

Main article: Subring

A subset S of R is a subring if it satisfies any of these equivalent conditions:

The addition and multiplication of R restrict to operations S × S → S that make S a ring with the same multiplicative identity as R.
1 ∈ S; and for all x, y in S, the elements xy, x + y, and −x are also in S.
S can be equipped with operations making it a ring such that the inclusion map S → R is a ring homomorphism.

For example, the ring Z of integers is a subring of the field of real numbers and also a subring of the ring of polynomials Z[X]. In both cases, Z contains 1, which is the multiplicative identity of the larger rings. On the other hand, the subset of even integers 2Z does not contain 1 and therefore does not qualify as a subring of Z. You could call 2Z a subrng, if you're into that sort of thing.

An intersection of subrings is a subring. Given a subset E of R, the smallest subring of R containing E is the intersection of all subrings of R containing E, called the subring generated by E.

For a ring R, its smallest subring is called the characteristic subring. It's generated by adding copies of 1 and –1. It's possible that n ⋅ 1 = 1 + 1 + ... + 1 (n times) could be zero. If n is the smallest positive integer for which this happens, n is the characteristic of R. In some rings, n ⋅ 1 is never zero for any positive integer n; these rings have characteristic zero.

Given a ring R, let Z(R) be the set of all elements x in R that commute with every other element: xy = yx for any y in R. Z(R) is a subring of R called the center. More generally, for a subset X of R, the set S of all elements in R that commute with every element in X is a subring of R called the centralizer of X. The center is the centralizer of the entire ring.

Ideal

Main article: Ideal (ring theory)

Let R be a ring. A left ideal of R is a nonempty subset I of R such that for any x, y in I and any r in R, the elements x + y and rx are in I. If RI denotes the R-span of I—that is, the set of finite sums r₁x₁ + ... + r_nx_n where r_i ∈ R and x_i ∈ I—then I is a left ideal if RI ⊆ I. Similarly, a right ideal is a subset I such that IR ⊆ I. A subset I is a two-sided ideal or simply an ideal if it's both a left and a right ideal. An ideal is like a black hole: multiply an element of the ideal by any element of the ring, and the result gets sucked back into the ideal.

If x is in R, then Rx and xR are left and right ideals, respectively, called the principal left and right ideals generated by x. The principal ideal RxR is written as (x). For example, the set of all multiples of 2 (positive and negative, plus 0) forms an ideal in the integers, generated by 2. In fact, every ideal in the ring of integers is principal.

A ring is called simple if it is nonzero and has no proper nonzero two-sided ideals. A commutative simple ring is precisely a field.

Rings are often studied by imposing conditions on their ideals. For instance, a ring with no strictly increasing infinite chain of left ideals is a left Noetherian ring. A ring with no strictly decreasing infinite chain of left ideals is a left Artinian ring. It is a surprising fact that a left Artinian ring is also left Noetherian (the Hopkins–Levitzki theorem). The integers, however, are a Noetherian ring that is not Artinian.

For commutative rings, ideals generalize the classical idea of divisibility. A proper ideal P of R is a prime ideal if for any elements x, y ∈ R, if xy ∈ P, then either x ∈ P or y ∈ P. This should sound familiar.

Homomorphism

Main article: Ring homomorphism

A homomorphism from a ring (R, +, ⋅) to a ring (S, ‡, ∗) is a function f from R to S that respects the ring operations. Specifically, for all a, b in R:

f(a + b) = f(a) ‡ f(b)
f(a ⋅ b) = f(a) ∗ f(b)
f(1_R) = 1_S

If you're slumming it with rngs, you drop the third condition. A ring homomorphism f is an isomorphism if there's an inverse homomorphism to f (i.e., a ring homomorphism that is an inverse function), or equivalently, if it's bijective.

Examples:

The function mapping each integer x to its remainder modulo 4 is a homomorphism from the ring Z to the quotient ring Z/4Z.
If u is a unit in a ring R, then R → R, x ↦ uxu⁻¹ is a ring homomorphism called an inner automorphism of R.
Let R be a commutative ring of prime characteristic p. Then x ↦ x^p is a ring endomorphism of R called the Frobenius homomorphism.
The Galois group of a field extension L/K is the set of all automorphisms of L that fix K.
For any ring R, there is a unique ring homomorphism Z → R and a unique ring homomorphism R → 0.
An epimorphism of rings need not be surjective. For example, the unique map Z → Q is an epimorphism.
An algebra homomorphism from a k-algebra to the endomorphism algebra of a vector space over k is a representation of the algebra.

The set of all elements mapped to 0 by a ring homomorphism f : R → S is the kernel of f. The kernel is a two-sided ideal of R. The image of f, however, is not always an ideal, but it is always a subring of S.

Quotient ring

Main article: Quotient ring

The notion of a quotient ring is analogous to that of a quotient group. Given a ring (R, +, ⋅) and a two-sided ideal I, you can view I as a subgroup of (R, +). The quotient ring R/I is the set of cosets of I with the operations:

(a + I) + (b + I) = (a + b) + I
(a + I) (b + I) = (ab) + I for all a, b in R. The ring R/I is also called a factor ring.

There's a canonical homomorphism p : R → R/I, given by x ↦ x + I. It is surjective and satisfies a universal property: if f : R → S is a ring homomorphism with f(I) = 0, then there exists a unique homomorphism f̄ : R/I → S such that f = f̄ ∘ p.

For any ring homomorphism f : R → S, this universal property, with I = ker f, yields a homomorphism f̄ : R/ker f → S that gives an isomorphism from R/ker f to the image of f.

Modules

Main article: Module (mathematics)

The concept of a module over a ring generalizes the concept of a vector space over a field. Instead of multiplying vectors by scalars from a field (scalar multiplication), you multiply them by elements of a ring. More precisely, for a ring R, an R-module M is an abelian group with an operation R × M → M that satisfies certain axioms. This operation is usually denoted by juxtaposition and called multiplication. For all a, b in R and all x, y in M:

M is an abelian group under addition.
a(x + y) = ax + ay
(a + b)x = ax + bx
1x = x
(ab)x = a(bx)

When the ring is noncommutative, these axioms define left modules. Right modules are defined similarly by writing xa instead of ax. This isn't just a notational quirk; the last axiom for right modules, x(ab) = (xa)b, becomes (ab)x = b(ax) if you try to use left multiplication for a right module.

Basic examples of modules include ideals and the ring itself. The theory of modules is vastly more complicated than that of vector spaces, mainly because modules are not characterized by a single invariant like dimension. Not all modules even have a basis.

Any ring homomorphism f : R → S induces a module structure: S becomes a left module over R via the multiplication rs = f(r)s. If R is commutative or if f(R) is in the center of S, then S is called an R-algebra. In particular, every ring is an algebra over the integers.

Constructions

Direct product

Main article: Direct product of rings

Given two rings R and S, their product R × S can be given a ring structure:

(r₁, s₁) + (r₂, s₂) = (r₁ + r₂, s₁ + s₂)
(r₁, s₁) ⋅ (r₂, s₂) = (r₁ ⋅ r₂, s₁ ⋅ s₂) for all r₁, r₂ in R and s₁, s₂ in S. This ring, with multiplicative identity (1, 1), is the direct product of R and S. The same construction works for any family of rings: if R_i are rings indexed by a set I, then ∏_i∈IR_i is a ring with componentwise operations.

If R is a commutative ring and a₁, ..., a_n are ideals such that a_i + a_j = (1) whenever i ≠ j, the Chinese remainder theorem provides a canonical ring isomorphism: R/(∩_i=1ⁿa_i) ≅ ∏_i=1ⁿ(R/a_i), via x mod ∩a_i ↦ (x mod a₁, ..., x mod a_n).

A finite direct product can also be seen as a direct sum of ideals.[36] Let R_i be rings, and consider the inclusions R_i → R = ∏R_i with images a_i. These a_i are ideals of R, and R = a₁ ⊕ ... ⊕ a_n as a direct sum of abelian groups. This can be understood through central idempotents. If R has such a decomposition, we can write 1 = e₁ + ... + e_n, where e_i ∈ a_i. These e_i are central idempotents that are orthogonal (e_ie_j = 0 for i ≠ j).

Polynomial ring

Main article: Polynomial ring

Given a symbol t (a variable) and a commutative ring R, the set of polynomials R[t] = {a_ntⁿ + a_n−1tⁿ⁻¹ + ... + a₁t + a₀ | n ≥ 0, a_j ∈ R} forms a commutative ring with the usual addition and multiplication, containing R as a subring. This is the polynomial ring over R. More generally, the set R[t₁, ..., t_n] of all polynomials in variables t₁, ..., t_n forms a commutative ring.

If R is an integral domain, then R[t] is also an integral domain; its field of fractions is the field of rational functions. If R is a Noetherian ring, so is R[t]. If R is a unique factorization domain, so is R[t]. Finally, R is a field if and only if R[t] is a principal ideal domain.

Let R ⊆ S be commutative rings. For an element x of S, there is a ring homomorphism R[t] → S given by f ↦ f(x) (this is substitution). The image of this map is denoted by R[x]; it is the subring of S generated by R and x.

The universal property of a polynomial ring states: given a ring homomorphism φ : R → S and an element x in S, there exists a unique ring homomorphism φ̄ : R[t] → S such that φ̄(t) = x and φ̄ restricts to φ.[37]

Another related construction is the formal power series ring R[[t]], consisting of series ∑₀^∞a_itⁱ. It doesn't have the universal property of a polynomial ring, as a series might not converge after substitution. Its main advantage is that it is a local ring.

Matrix ring and endomorphism ring

Main articles: Matrix ring and Endomorphism ring

Let R be a ring. The set of all square matrices of size n with entries in R forms a ring with entry-wise addition and standard matrix multiplication, denoted M_n(R). Given a right R-module U, the set of all R-linear maps from U to itself forms a ring with function addition and composition of functions as multiplication. This is the endomorphism ring of U, denoted End_R(U).

A matrix ring can be seen as an endomorphism ring: End_R(Rⁿ) ≅ M_n(R).

Schur's lemma states that if U is a simple right R-module, then End_R(U) is a division ring.[40] The Artin–Wedderburn theorem says any semisimple ring is a finite product of matrix rings over division rings.

A ring R and the matrix ring M_n(R) are Morita equivalent: the category of right modules of R is equivalent to the category of right modules over M_n(R).[39] This means their module theories are essentially the same.

Limits and colimits of rings

Let R_i be a sequence of rings where R_i is a subring of R_i+1 for all i. The union (or filtered colimit) of the R_i is the ring lim_→R_i. An example is a polynomial ring in infinitely many variables: R[t₁, t₂, ...] = lim_→R[t₁, ..., t_m]. Any commutative ring is the colimit of its finitely generated subrings.

A projective limit (or filtered limit) of rings is defined dually. Given a family of rings R_i and homomorphisms R_j → R_i for j ≥ i, the projective limit lim_←R_i is the subring of ∏R_i consisting of sequences (x_n) that are compatible with the homomorphisms.

Localization

Localization generalizes the construction of the field of fractions of an integral domain. Given a ring R and a subset S, there exists a ring R[S⁻¹] and a homomorphism R → R[S⁻¹] that "inverts" S, mapping elements of S to units. This construction is universal: any other homomorphism from R that inverts S factors uniquely through R[S⁻¹].[41]

Localization is often applied to a commutative ring R with respect to the complement of a prime ideal p. In this case, S = R − p, and the localization is denoted R_p. This R_p is a local ring with the maximal ideal pR_p, hence the name "localization."

Completion

Let R be a commutative ring and I be an ideal. The completion of R at I is the projective limit R̂ = lim_←R/Iⁿ. This is a commutative ring. The construction is particularly useful when I is a maximal ideal.

The quintessential example is the completion of Z at the ideal (p) for a prime p. This yields the ring of p-adic integers, denoted Zp. A complete ring often has a much simpler structure than a general commutative ring, as described by the Cohen structure theorem.

Rings with generators and relations

The most general way to build a ring is to specify generators and relations. Start with a free ring F on a set of symbols X (polynomials with integer coefficients in noncommuting variables). Then, impose relations by taking a quotient. If E is a subset of F, the quotient of F by the ideal generated by E is the ring with generators X and relations E. Every ring can be represented as a quotient of a free ring.[46]

Special kinds of rings

Domains

A nonzero ring with no nonzero zero-divisors is a domain. A commutative domain is an integral domain. The most important are principal ideal domains (PIDs), where every ideal is principal, and fields. A broader class is the unique factorization domain (UFD), where every nonunit element is a product of prime elements. A fundamental question in algebraic number theory concerns the degree to which the ring of integers in a number field fails to be a PID.

The following chain of class inclusions describes the hierarchy: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields

Division ring

A division ring is a ring where every non-zero element is a unit. A commutative division ring is a field. The classic example of a non-field division ring is the ring of quaternions. Every finite domain is a field, a result known as Wedderburn's little theorem.

Semisimple rings

Main article: Semisimple module

A semisimple module is a direct sum of simple modules. A semisimple ring is a ring that is semisimple as a left (or right) module over itself. By the Artin–Wedderburn theorem, a ring R is semisimple if and only if it is a finite direct product ∏_i=1^r M_{n_i}(D_i), where each D_i is a division ring. A division ring is semisimple. For a field k and finite group G, the group ring kG is semisimple if and only if the characteristic of k does not divide the order of G (Maschke's theorem).

Central simple algebra and Brauer group

Main article: Central simple algebra

For a field k, a k-algebra is central if its center is k and simple if it is a simple ring. Two central simple algebras A and B are similar if A ⊗_k k_n ≈ B ⊗_k k_m for some integers n and m.[49] This equivalence relation gives rise to the Brauer group of k, denoted Br(k), an abelian group whose elements are the similarity classes.

Valuation ring

Main article: Valuation ring

If K is a field, a valuation v is a group homomorphism from K^∗ to a totally ordered abelian group G such that v(f + g) ≥ min{v(f), v(g)}. The valuation ring of v is the subring of K consisting of zero and all nonzero f such that v(f) ≥ 0.

Rings with extra structure

A ring is an abelian group with additional structure. Other mathematical objects can be seen as rings with even more structure.

An associative algebra is a ring that is also a vector space over a field k, where scalar multiplication is compatible with ring multiplication.
A topological ring is a ring whose set of elements is given a topology that makes addition and multiplication continuous.
A λ-ring is a commutative ring R with operations λⁿ: R → R that behave like nth exterior powers.
A totally ordered ring is a ring with a total ordering compatible with its operations.

Some examples of the ubiquity of rings

Various mathematical objects can be analyzed through some associated ring. It's a distressingly common tactic.

Cohomology ring of a topological space: To any topological space X, one can associate its integral cohomology ring H^∗(X, Z), a graded ring. This structure is fundamental to characteristic classes, intersection theory, and Schubert calculus.
Burnside ring of a group: To any group, one associates its Burnside ring, which describes the ways the group can act on a finite set.
Representation ring of a group ring: To any group ring or Hopf algebra, one associates its representation ring. The additive group is based on indecomposable modules, and multiplication is the tensor product.
Function field of an irreducible algebraic variety: To any irreducible algebraic variety is associated its function field. The study of algebraic geometry heavily uses commutative algebra to translate geometric concepts into ring-theoretic properties.
Face ring of a simplicial complex: Every simplicial complex has an associated face ring, or Stanley–Reisner ring, which reflects its combinatorial properties and is crucial in algebraic combinatorics.

Category-theoretic description

Generalization

Because mathematicians can never leave well enough alone, structures more general than rings have been defined by weakening or dropping some of the ring axioms.

Rng: A rng is a ring without the assumption of a multiplicative identity.[52]
Nonassociative ring: A nonassociative ring satisfies all ring axioms except for associative multiplication and the existence of a multiplicative identity. A Lie algebra is a notable example.
Semiring: A semiring (or rig) weakens the assumption that (R, +) is an abelian group to it being a commutative monoid and adds the axiom that 0 ⋅ a = a ⋅ 0 = 0. An example is the non-negative integers {0, 1, 2, ...} with ordinary addition and multiplication.

Other ring-like objects

Ring object in a category: In a category C with finite products, a ring object is an object R with morphisms for addition, multiplication, and identities that satisfy the usual ring axioms.
Ring scheme: In algebraic geometry, a ring scheme over a base scheme S is a ring object in the category of S-schemes. An example is the ring scheme W_n over Spec Z, which for any commutative ring A returns the ring W_n(A) of p-isotypic Witt vectors of length n over A.[53]
Ring spectrum: In algebraic topology, a ring spectrum is a spectrum X with a multiplication and a unit map from the sphere spectrum S, such that the ring axiom diagrams commute up to homotopy.