- 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.

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