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 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, 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 real square matrices with, 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 Dedekind, Hilbert, Fraenkel, and 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:

Definition

A ring is a set equipped with two binary operations + and ⋅ satisfying the following three sets of axioms, called the ring axioms:
  1. is an abelian group under addition, meaning that:
  2. * for all in .
  3. * for all in .
  4. * There is an element in such that for all in .
  5. * For each in there exists in such that .
  6. is a monoid under multiplication, meaning that:
  7. * for all in .
  8. * There is an element in such that and for all in .
  9. Multiplication is distributive with respect to addition, meaning that:
  10. * for all in .
  11. * for all in .
In notation, the multiplication symbol is often omitted, in which case is written as.

Variations on terminology

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 |" 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 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: need not necessarily equal. Rings that also satisfy commutativity for multiplication 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 "", and does not work in a rng.
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".

Illustration

The most familiar example of a ring is the set of all integers consisting of the numbers
The axioms of a ring are modeled on familiar properties of addition and multiplication of integers.

Some properties

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 in a ring, one has and.
  • If in a ring , then has only one element, and is called the zero ring.
  • If a ring contains the zero ring as a subring, then itself is the zero ring.
  • The binomial formula holds for any and satisfying.

    Example: Integers modulo 4

Equip the set with the following operations:
  • The sum in is the remainder when the integer is divided by . For example, and
  • The product in is the remainder when the integer is divided by. For example, and
Then is a ring: each axiom follows from the corresponding axiom for If is an integer, the remainder of when divided by may be considered as an element of and this element is often denoted by "" or which is consistent with the notation for. The additive inverse of any in is For example,

Example: 2-by-2 matrices

The set of 2-by-2 square matrices with entries in a field is
With the operations of matrix addition and matrix multiplication, satisfies the above ring axioms. The element is the multiplicative identity of the ring. If and then while this example shows that the ring is noncommutative.
More generally, for any ring, commutative or not, and any nonnegative integer, the square matrices with entries in form a ring; see Matrix ring.

History

Dedekind

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" 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.

Hilbert

The term "Zahlring" 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. 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 then:
and so on; in general, is going to be an integral linear combination of,, and.

Fraenkel and Noether

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 and developed the foundations of commutative ring theory in her paper Idealtheorie in Ringbereichen.

Multiplicative identity and the term "ring"

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 for a "ring". Starting in the 1960s, it became increasingly common to see books including the existence of 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. 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, if one requires all rings to have a, 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.

    Basic examples

Commutative rings