- Unit (ring theory)
In

mathematics , a**unit**in a (unital ) ring "R" is an invertible element of "R", i.e. an element "u" such that there is a "v" in "R" with :"uv" = "vu" = 1_{"R"}, where 1_{"R"}is the multiplicativeidentity element .That is, "u" is an "invertible" element of the multiplicative

monoid of "R". If $0\; e\; 1$ in the ring, then $0$ is not a unit.Unfortunately, the term "unit" is also used to refer to the identity element 1

_{"R"}of the ring, in expressions like "ring with a unit" or "unit ring ", and also e.g. "'unit' matrix". (For this reason, some authors call 1_{R}"unity", and say that "R" is a "ring with unity" rather than "ring with a unit". Note also that the term "unit matrix " more usually denotes a matrix with all diagonal elements equal to one, and all other elements equal to zero.)If $0\; e\; 1$ and the sum of any two non-units is not a unit, then the ring is a

local ring .**Group of units**The units of "R" form a group "U"("R") under multiplication, the

**group of units**of "R". The group of units "U"("R") is sometimes also denoted "R"^{*}or "R"^{×}.In a commutative unital ring "R", the group of units "U"("R") acts on "R" via multiplication. The orbits of this action are called sets of "associates"; in other words, there is an

equivalence relation ~ on "R" called "associatedness" such that:"r" ~ "s"

means that there is a unit "u" with "r" = "us".

One can check that "U" is a

functor from thecategory of rings to thecategory of groups : everyring homomorphism "f" : "R" → "S" induces agroup homomorphism "U"("f") : "U"("R") → "U"("S"), since "f" maps units to units. This functor has aleft adjoint which is the integralgroup ring construction.In an

integral domain thecardinality of an equivalence class of associates is the same as that of "U"("R").A ring "R" is a

division ring if and only if "R"^{*}= "R" {0}.**Examples*** In the

ring of integers ,**Z**, the units are ±1. The associates are pairs "n" and −"n".* In the ring of integers modulo "n",

**Z**/"n**"Z**, the units are the congruence classes (mod "n") which arecoprime to "n". They constitute the multiplicative group of integers (mod "n").* Any

root of unity is a unit in any unital ring "R". (If "r" is a root of unity, and "r"^{"n"}= 1, then "r"^{−1}= "r"^{"n" − 1}is also an element of "R" by closure under multiplication.) Inalgebraic number theory ,Dirichlet's unit theorem shows the existence of many units in most rings ofalgebraic integer s. For example, we have (√5 + 2)(√5 − 2) = 1.* In the ring "M"("n",

**F**) of "n"×"n" matrices over some field**F**the units are exactly the invertible matrices.

*Wikimedia Foundation.
2010.*

### Look at other dictionaries:

**Unit ring**— In mathematics, a unit ring or ring with a unit is a unital ring, i.e. a ring R with a (multiplicative) unit element, denoted by 1 R or simply 1 if there is no risk of confusion. Alternative definitions of a ring Some authors (such as Herstein)… … Wikipedia**Ideal (ring theory)**— In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like even number or multiple of 3 . For instance, in… … Wikipedia**Divisibility (ring theory)**— In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. Please see the page about divisors for this simplest example. With the development of abstract rings, of which the integers are the… … Wikipedia**Domain (ring theory)**— In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0.[1] That is, it is a ring which has no left or right zero divisors. (Sometimes such a ring is… … Wikipedia**Glossary of ring theory**— Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. Contents 1 Definition of a ring 2 Types of… … Wikipedia**Fundamental unit (number theory)**— In algebraic number theory, a fundamental unit is a generator for the torsion free unit group of the ring of integers of a number field, when that group is infinite cyclic. See also Dirichlet s unit theorem.For rings of the form mathbb{Z} [sqrt… … Wikipedia**Unit**— may refer to:In mathematics: * Unit vector, a vector with length equal to 1 * Unit circle, the circle with radius equal to 1, centered at the origin * Unit interval, the interval of all real numbers between 0 and 1 * Imaginary unit, i , whose… … Wikipedia**Ring (mathematics)**— This article is about algebraic structures. For geometric rings, see Annulus (mathematics). For the set theory concept, see Ring of sets. Polynomials, represented here by curves, form a ring under addition and multiplication. In mathematics, a… … Wikipedia**Ring homomorphism**— In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication. More precisely, if R and S are rings, then a ring homomorphism is a function f : R → S such that … Wikipedia**ring**— ring1 ringless, adj. ringlike, adj. /ring/, n., v., ringed, ringing. n. 1. a typically circular band of metal or other durable material, esp. one of gold or other precious metal, often set with gems, for wearing on the finger as an ornament, a… … Universalium