Ring (mathematics)

In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. 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.

This article is about the algebraic structure. For other uses in mathematics, see Ring (disambiguation) § Mathematics.

Formally, a ring is a set endowed with two binary operations called addition and multiplication such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors define rings without requiring a multiplicative identity and instead call the structure defined above a ring with identity. See § Variations on the definition.)

Whether a ring is commutative has profound implications on its behavior. 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. The simplest commutative rings are those that admit division by non-zero elements; such rings are called fields.

Examples of commutative rings include the set of integers with their standard addition and multiplication, the set of polynomials with their addition and multiplication, 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 $n \times n$ real square matrices with $n \geq 2$ , 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 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.

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 $x 0 = 0 = 0 x$ (zero is an with respect to multiplication) and $(-1) x = - x$ .

absorbing element

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.

[6]

The holds for any $x$ and $y$ satisfying $xy = yx$ .

binomial formula

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

polynomials

The ring of , the integral closure of $\mathbb {Z}$ in a quadratic extension of $\mathbb {Q} .$ It is a subring of the ring of all algebraic integers.

quadratic integers

The ring of ${\widehat {\mathbb {Z} }},$ the (infinite) product of the rings of $p$ -adic integers $\mathbb {Z} _{p}$ over all prime numbers $p$ .

profinite integers

The , the ring generated by Hecke operators.

Hecke ring

If $S$ is a set, then the 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.

power set

Basic concepts[edit]

Products and powers[edit]

For each nonnegative integer $n$ , given a sequence $(a_{1},\dots ,a_{n})$ of $n$ elements of $R$ , one can define the product $P_{n}=\prod _{i=1}^{n}a_{i}$ recursively: let $P 0 = 1$ and let $P m = P m -1 a m$ for $1 \leq m \leq n$ .

As a special case, one can define nonnegative integer powers of an element $a$ of a ring: $a 0 = 1$ and $a n = a n -1 a$ for $n \geq 1$ . Then $a m + n = a m a n$ for all $m, n \geq 0$ .

Elements in a ring[edit]

A left zero divisor of a ring $R$ is an element $a$ in the ring such that there exists a nonzero element $b$ of $R$ such that $ab = 0$ .^[d] A right zero divisor is defined similarly.

A nilpotent element is an element $a$ such that $a n = 0$ for some $n > 0$ . One example of a nilpotent element is a nilpotent matrix. A nilpotent element in a nonzero ring is necessarily a zero divisor.

An idempotent $e$ is an element such that $e 2 = e$ . One example of an idempotent element is a projection in linear algebra.

A unit is an element $a$ having a multiplicative inverse; in this case the inverse is unique, and is denoted by $a -1$ . The set of units of a ring is a group under ring multiplication; this group is denoted by $R \times$ or $R *$ or $U (R)$ . For example, if $R$ is the ring of all square matrices of size $n$ over a field, then $R \times$ consists of the set of all invertible matrices of size $n$ , and is called the general linear group.

A polynomial ring in infinitely many variables: $R[t_{1},t_{2},\cdots ]=\varinjlim R[t_{1},t_{2},\cdots ,t_{m}].$

The of finite fields of the same characteristic ${\overline {\mathbf {F} }}_{p}=\varinjlim \mathbf {F} _{p^{m}}.$

algebraic closure

The field of over a field $k$ : $k(\!(t)\!)=\varinjlim t^{-m}k[\![t]\!]$ (it is the field of fractions of the formal power series ring $k[\![t]\!].$ )

formal Laurent series

The over a field $k$ is $\varinjlim k[U]$ where the limit runs over all the coordinate rings $k [U]$ of nonempty open subsets $U$ (more succinctly it is the stalk of the structure sheaf at the generic point.)

function field of an algebraic variety

Special kinds of rings[edit]

Domains[edit]

A nonzero ring with no nonzero zero-divisors is called a domain. A commutative domain is called an integral domain. The most important integral domains are principal ideal domains, PIDs for short, and fields. A principal ideal domain is an integral domain in which every ideal is principal. An important class of integral domains that contain a PID is a unique factorization domain (UFD), an integral domain in which every nonunit element is a product of prime elements (an element is prime if it generates a prime ideal.) The fundamental question in algebraic number theory is on the extent to which the ring of (generalized) integers in a number field, where an "ideal" admits prime factorization, fails to be a PID.

Among theorems concerning a PID, the most important one is the structure theorem for finitely generated modules over a principal ideal domain. The theorem may be illustrated by the following application to linear algebra.^[47] Let $V$ be a finite-dimensional vector space over a field $k$ and $f : V \to V$ a linear map with minimal polynomial $q$ . Then, since $k [t]$ is a unique factorization domain, $q$ factors into powers of distinct irreducible polynomials (that is, prime elements):

An is a ring that is also a vector space over a field $n$ such that the scalar multiplication is compatible with the ring multiplication. For instance, the set of $n$ -by- $n$ matrices over the real field $\mathbb {R}$ has dimension $n 2$ as a real vector space.

associative algebra

A ring $R$ is a if its set of elements $R$ is given a topology which makes the addition map ( $+:R\times R\to R$ ) and the multiplication map $\cdot : R \times R \to R$ to be both continuous as maps between topological spaces (where $X \times X$ inherits the product topology or any other product in the category). For example, $n$ -by- $n$ matrices over the real numbers could be given either the Euclidean topology, or the Zariski topology, and in either case one would obtain a topological ring.

topological ring

A is a commutative ring $R$ together with operations $λ n : R \to R$ that are like $n$ th exterior powers:
$\lambda ^{n}(x+y)=\sum _{0}^{n}\lambda ^{i}(x)\lambda ^{n-i}(y).$

λ-ring

A ring may be viewed as an abelian group (by using the addition operation), with extra structure: namely, ring multiplication. In the same way, there are other mathematical objects which may be considered as rings with extra structure. For example:

the non-negative integers $\{0,1,2,\ldots \}$ with ordinary addition and multiplication;

the .

tropical semiring

Other ring-like objects[edit]

Ring object in a category[edit]

Let $C$ be a category with finite products. Let pt denote a terminal object of $C$ (an empty product). A ring object in $C$ is an object $R$ equipped with morphisms $R\times R\;{\stackrel {a}{\to }}\,R$ (addition), $R\times R\;{\stackrel {m}{\to }}\,R$ (multiplication), $\operatorname {pt} {\stackrel {0}{\to }}\,R$ (additive identity), $R\;{\stackrel {i}{\to }}\,R$ (additive inverse), and $\operatorname {pt} {\stackrel {1}{\to }}\,R$ (multiplicative identity) satisfying the usual ring axioms. Equivalently, a ring object is an object $R$ equipped with a factorization of its functor of points $h_{R}=\operatorname {Hom} (-,R):C^{\operatorname {op} }\to \mathbf {Sets}$ through the category of rings: $C^{\operatorname {op} }\to \mathbf {Rings} {\stackrel {\textrm {forgetful}}{\longrightarrow }}\mathbf {Sets} .$

Ring scheme[edit]

In algebraic geometry, a ring scheme over a base scheme $S$ is a ring object in the category of $S$ -schemes. One example is the ring scheme $W n$ over $\operatorname {Spec} \mathbb {Z}$ , 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[edit]

In algebraic topology, a ring spectrum is a spectrum $X$ together with a multiplication $\mu :X\wedge X\to X$ and a unit map $S \to X$ from the sphere spectrum $S$ , such that the ring axiom diagrams commute up to homotopy. In practice, it is common to define a ring spectrum as a monoid object in a good category of spectra such as the category of symmetric spectra.