Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set with two binary operations typically called addition and multiplication and denoted like addition and multiplication of integers. They work similarly to integer addition and multiplication, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

More formally, a ring is a set that is endowed with two binary operations (addition and multiplication) such that the ring is an abelian group with respect to addition. The multiplication is associative, is distributive over the addition operation, and has a multiplicative identity element. Some authors apply the term ring to a further generalization, often called a rng, that omits the requirement for a multiplicative identity, and instead call the structure defined above a ring with identity.

A commutative ring is a ring with a commutative multiplication. This property has profound implications on ring properties. Commutative algebra, the theory of commutative rings, is a major branch of ring theory. Its development has been greatly influenced by problems and ideas of algebraic number theory and algebraic geometry. In turn, commutative algebra is a fundamental tool in these branches of mathematics.

Examples of commutative rings include every field (such as the real or complex numbers), the integers, the polynomials in one or several variables with coefficients in another ring, the coordinate ring of an affine algebraic variety, and the ring of integers of a number field. Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators, and cohomology rings in topology.

The conceptualization of rings spanned the 1870s to the 1920s, with key contributions by Richard Dedekind, David Hilbert, Abraham Fraenkel, and Emmy Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, and of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. They later proved useful in other branches of mathematics such as geometry and analysis.

Rings appear in the following 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

A ring is a set R equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms:

1. R is an abelian group under addition, meaning that:
   • (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative).
   • a + b = b + a for all a, b in R (that is, + is commutative).
   • There is an element 0 in R such that a + 0 = a for all a in R (that is, 0 is an additive identity).
   • For each a in R there exists −a in R such that a + (−a) = 0 (that is, −a is the additive inverse of a).
2. R is a monoid under multiplication, meaning that:
   • (a · b) · c = a · (b · c) for all a, b, c in R (that is, ⋅ is associative).
   • There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R (that is, 1 is a multiplicative identity).
3. Multiplication is distributive with respect to addition, meaning that:
   • 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).

In notation, the multiplication symbol · is often omitted, in which case a · b is written as ab.

In the terminology of this article, a ring is defined to have a multiplicative identity, while a structure with the same axiomatic definition but without the requirement for a multiplicative identity is instead called a "rng" (IPA: /rʊŋ/) with a missing "i". For example, the set of even integers with the usual + and ⋅ is a rng, but not a ring. As explained in § History below, many authors apply the term "ring" without requiring a multiplicative identity.

Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not necessarily equal ba. Rings that also satisfy commutativity for multiplication (such as the ring of integers) are called commutative rings. Books on commutative algebra or algebraic geometry often adopt the convention that ring means commutative ring, to simplify terminology.

In a ring, multiplicative inverses are not required to exist. A nonzero ring in which every nonzero element has a multiplicative inverse is called a division ring and a commutative division ring is called a field.

The additive group of a ring is the underlying set equipped with only the operation of addition. Although the definition requires that the additive group be abelian, this can be inferred from the other ring axioms. The proof makes use of the "1", and does not work in a rng. (For a rng, omitting the axiom of commutativity of addition leaves it inferable from the remaining rng assumptions only for elements that are products: ab + cd = cd + ab.)

Some authors use the term "ring" to refer to structures in which there is no requirement for multiplication to be associative; see the nonassociative ring subsection below. For these authors, every algebra is a "ring".

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

\(\dots,-5,-4,-3,-2,-1,0,1,2,3,4,5,\dots\)

The axioms of a ring are modeled on familiar properties of addition and multiplication of integers.

Some basic properties of a ring follow immediately from the axioms:

• The additive identity is unique.
• The additive inverse of each element is unique.
• The multiplicative identity is unique.
• For any element x in a ring R, one has x0 = 0 = 0x (zero is an absorbing element with respect to multiplication) and (–1)x = –x.
• If 0 = 1 in a ring R (or more generally, 0 is a unit element), then R has only one element, and is called the zero ring.
• If a ring R contains the zero ring as a subring, then R itself is the zero ring.
• The binomial formula holds for any x and y satisfying xy = yx.

Equip the set \(\Z /4\Z = \left\{\overline{0}, \overline{1}, \overline{2}, \overline{3}\right\}\) with the following operations:

• The sum \(\overline{x} + \overline{y}\) in ⁠\(\Z/4\Z\)⁠ is the remainder when the integer x + y is divided by 4 (as x + y is always smaller than 8, this remainder is either x + y or x + y − 4). For example, \(\overline{2} + \overline{3} = \overline{1}\) and \(\overline{3} + \overline{3} = \overline{2}.\)
• The product \(\overline{x} \cdot \overline{y}\) in ⁠\(\Z/4\Z\)⁠ is the remainder when the integer xy is divided by 4. For example, \(\overline{2} \cdot \overline{3} = \overline{2}\) and \(\overline{3} \cdot \overline{3} = \overline{1}.\)

Then ⁠\(\Z/4\Z\)⁠ is a ring: each axiom follows from the corresponding axiom for ⁠\(\Z.\)⁠ If x is an integer, the remainder of x when divided by 4 may be considered as an element of ⁠\(\Z/4\Z,\)⁠ and this element is often denoted by "x mod 4" or \(\overline x,\) which is consistent with the notation for 0, 1, 2, 3. The additive inverse of any \(\overline x\) in ⁠\(\Z/4\Z\)⁠ is \(-\overline x=\overline{-x}.\) For example, \(-\overline{3} = \overline{-3} = \overline{1}.\)

The set of 2-by-2 square matrices with entries in a field F is

\(\operatorname{M}_2(F) = \left\{ \left.\begin{pmatrix} a & b \\ c & d \end{pmatrix} \right|\ a, b, c, d \in F \right\}.\)

With the operations of matrix addition and matrix multiplication, \(\operatorname{M}_2(F)\) satisfies the above ring axioms. The element \(\left( \begin{smallmatrix} 1 & 0 \\ 0 & 1 \end{smallmatrix}\right)\) is the multiplicative identity of the ring. If \(A = \left( \begin{smallmatrix} 0 & 1 \\ 1 & 0 \end{smallmatrix} \right)\) and \(B = \left( \begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix} \right),\) then \(AB = \left( \begin{smallmatrix} 0 & 0 \\ 0 & 1 \end{smallmatrix} \right)\) while \(BA = \left( \begin{smallmatrix} 1 & 0 \\ 0 & 0 \end{smallmatrix} \right);\) this example shows that the ring is noncommutative.

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.

The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. In 1871, Richard Dedekind defined the concept of the ring of integers of a number field. In this context, he introduced the terms "ideal" (inspired by Ernst Kummer's notion of ideal number) and "module" and studied their properties. Dedekind did not use the term "ring" and did not define the concept of a ring in a general setting.

The term "Zahlring" (number ring) was coined by David Hilbert in 1892 and published in 1897. According to Harvey Cohn, Hilbert used the term for a ring that had the property of "circling directly back" to an element of itself (in the sense of an equivalence). Specifically, in a ring of algebraic integers, all high powers of an algebraic integer can be written as an integral combination of a fixed set of lower powers, and thus the powers "cycle back". For instance, if a − 4a + 1 = 0 then:

\(\begin{align} a^3 &= 4a-1, \\ a^4 &= 4a^2-a, \\ a^5 &= -a^2+16a-4, \\ a^6 &= 16a^2-8a+1, \\ a^7 &= -8a^2+65a-16, \\ \vdots \ & \qquad \vdots \end{align}\)

and so on; in general, a is going to be an integral linear combination of 1, a, and a.

The first axiomatic definition of a ring was given by Abraham Fraenkel in 1915, but his axioms were stricter than those in the modern definition. For instance, he required every non-zero-divisor to have a multiplicative inverse. In 1921, Emmy Noether gave a modern axiomatic definition of commutative rings (with and without 1) and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.

Fraenkel applied the term "ring" to structures with axioms that included a multiplicative identity, whereas Noether applied it to structures that did not.

Most or all books on algebra up to around 1960 followed Noether's convention of not requiring a 1 for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of "ring", especially in advanced books by notable authors such as Artin, Bourbaki, Eisenbud, and Lang. There are also books published as late as 2022 that use the term without the requirement for a 1. Likewise, the Encyclopedia of Mathematics does not require unit elements in rings. In a research article, the authors often specify which definition of ring they use in the beginning of that article.

Gardner and Wiegandt assert that, when dealing with several objects in the category of rings (as opposed to working with a fixed ring), if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and 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." Poonen makes the counterargument that the natural notion for rings would be the direct product rather than the direct sum. However, his main argument is that rings without a multiplicative identity are not totally associative, in the sense that they do not contain the product of any finite sequence of ring elements, including the empty sequence.

Authors who follow either convention for the use of the term "ring" may use one of the following terms to refer to objects satisfying 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", or "ring with 1".
• to omit a requirement for a multiplicative identity: "rng" or "pseudo-ring", although the latter may be confusing because it also has other meanings.

• The prototypical example is the ring of integers with the two operations of addition and multiplication.
• The rational, real and complex numbers are commutative rings of a type called fields.
• A unital associative algebra over a commutative ring R is itself a ring as well as an R-module. Some examples:
  • The algebra R[X] of polynomials with coefficients in R.
  • The algebra \(R[[X_1, \dots, X_n]]\) 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 of functions.
  • Let X be a set, and let R be a ring. Then the set of all functions from X to R forms a ring, which is commutative if R is commutative.
• The ring of quadratic integers, the integral closure of ⁠\(\Z\)⁠ in a quadratic extension of ⁠\(\Q.\)⁠ It is a subring of the ring of all algebraic integers.
• The ring of profinite integers ⁠\(\widehat\Z,\)⁠ the (infinite) product of the rings of p-adic integers ⁠\(\Z _p\)⁠ over all prime numbers p.
• The Hecke ring, the ring generated by Hecke operators.
• If S is a set, then the power set of S becomes a ring if we define addition to be the symmetric difference of sets and multiplication to be intersection. This is an example of a Boolean ring.

• For any ring R and any natural number n, the set of all square n-by-n matrices with entries from R, forms a ring with matrix addition and matrix multiplication as operations. For n = 1, this matrix ring is isomorphic to R itself. For n > 1 (and R not the zero ring), this matrix ring is noncommutative.
• If G is an abelian group, then the endomorphisms of G form a ring, the endomorphism ring End(G) of G. The operations in this ring are addition and composition of endomorphisms. More generally, if V is a left module over a ring R, then the set of all R-linear maps forms a ring, also called the endomorphism ring and denoted by End_R(V).
• The endomorphism ring of an elliptic curve. It is a commutative ring if the elliptic 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 having G as basis. Multiplication is defined by the rules that the elements of G commute with the elements of R and multiply together as they do in the group G.
• The ring of differential operators (depending on the context). In fact, many rings that appear in analysis are noncommutative. For example, most Banach algebras are noncommutative.

• The set of natural numbers ⁠\(\N\)⁠ with the usual operations is not a ring, since ⁠\((\N, +)\)⁠ is not even a group (not all the elements are invertible with respect to addition, for instance, there is no natural number which can be added to 3 to get 0 as a result). There is a natural way to enlarge it to a ring, by including negative numbers to produce the ring of integers ⁠\(\Z.\)⁠ The natural numbers (including 0) form an algebraic structure known as a semiring (which has all of the axioms of a ring excluding that of an additive inverse).
• Let R be the set of all continuous functions on the real line that vanish outside a bounded interval that depends on the function, with addition as usual but with multiplication defined as convolution: \[(f * g)(x) = \int_{-\infty}^\infty f(y)g(x - y) \, dy.\] Then R is a rng, but not a ring: the Dirac delta function has the property of a multiplicative identity, but it is not a function and hence is not an element of R.