Unit (ring theory)


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 multiplicative identity 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 1R "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 the category of rings to the category of groups: every ring homomorphism "f" : "R" → "S" induces a group homomorphism "U"("f") : "U"("R") → "U"("S"), since "f" maps units to units. This functor has a left adjoint which is the integral group ring construction.

In an integral domain the cardinality 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 are coprime 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.) In algebraic number theory, Dirichlet's unit theorem shows the existence of many units in most rings of algebraic integers. 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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