Unit (ring theory)
Encyclopedia
In mathematics
, an invertible element or a unit in a (unital) ring
R refers to any element u that has an inverse element
in the multiplicative monoid
of R, i.e. such element v that
The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group
for this operation. It never contains the element 0 (except in the case of the trivial ring), and is therefore not closed under addition; its complement
however might be a group under addition, which happens if and only if the ring is a local ring
.
Unfortunately, the term unit is also used to refer to the identity element 1R 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".)
U(R) under multiplication, the group of units of R. Other common notations for U(R) are R*, R×, and E(R) (for the German term Einheit).
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
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}.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an invertible element or a unit in a (unital) ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
R refers to any element u that has an inverse element
Inverse element
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
in the multiplicative monoid
Monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
of R, i.e. such element v that
- uv = vu = 1R, where 1R is the multiplicative identity elementIdentity elementIn mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
.
The set of units of any ring is closed under multiplication (the product of two units is again a unit), and forms a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
for this operation. It never contains the element 0 (except in the case of the trivial ring), and is therefore not closed under addition; its complement
Complement (set theory)
In set theory, a complement of a set A refers to things not in , A. The relative complement of A with respect to a set B, is the set of elements in B but not in A...
however might be a group under addition, which happens if and only if the ring is a local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
.
Unfortunately, the term unit is also used to refer to the identity element 1R of the ring, in expressions like ring with a unit or unit ring
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 1R or simply 1 if there is no risk of confusion.- Alternative definitions of a ring :...
, and also e.g. 'unit' matrix
Identity matrix
In linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
. (For this reason, some authors call 1R "unity", and say that R is a "ring with unity" rather than "ring with a unit".)
Group of units
The units of R form a groupGroup (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
U(R) under multiplication, the group of units of R. Other common notations for U(R) are R*, R×, and E(R) (for the German term Einheit).
In a commutative unital ring R, the group of units U(R) acts
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on R via multiplication. The orbits of this action are called sets of associates; in other words, there is an equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
~ 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
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
from the category of rings
Category of rings
In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings and whose morphisms are ring homomorphisms...
to the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
: every ring homomorphism
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....
f : R → S induces a group homomorphism
Group homomorphism
In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...
U(f) : U(R) → U(S), since f maps units to units. This functor has a left adjoint which is the integral group ring
Group ring
In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is one-to-one with the given group. As a ring, its addition law is that of the free...
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
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...
if and only if
If and only if
In logic and related fields such as mathematics and philosophy, if and only if is a biconditional logical connective between statements....
R* = R \ {0}.
Examples
- In the ring of integersRing of integersIn mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
, Z, the units are ±1. The associates are pairs n and −n. - In the ring of integers modulo nModular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
, Z/nZ, the units are the congruence classes (mod n) which are coprimeCoprimeIn number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
to n. They constitute the multiplicative group of integers (mod n)Multiplicative group of integers modulo nIn modular arithmetic the set of congruence classes relatively prime to the modulus n form a group under multiplication called the multiplicative group of integers modulo n. It is also called the group of primitive residue classes modulo n. In the theory of rings, a branch of abstract algebra, it...
. - Any root of unityRoot of unityIn mathematics, a root of unity, or de Moivre number, is any complex number that equals 1 when raised to some integer power n. Roots of unity are used in many branches of mathematics, and are especially important in number theory, the theory of group characters, field theory, and the discrete...
is a unit in any unital ring R. (If r is a root of unity, and rn = 1, then r−1 = rn − 1 is also an element of R by closure under multiplication.) - If R is the ring of integersRing of integersIn mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
in a number field, Dirichlet's unit theoremDirichlet's unit theoremIn mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Gustav Lejeune Dirichlet. It determines the rank of the group of units in the ring OK of algebraic integers of a number field K...
states that the group of units of R is a finitely generated abelian group. For example, we have (√5 + 2)(√5 − 2) = 1 in the ring of integers of Q[√5], and in fact the unit group is infinite in this case. In general, the unit group of a real quadratic field is always infinite (of rank 1). - In the ring M(n,F) of n×n matricesMatrix (mathematics)In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
over a fieldField (mathematics)In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
F, the units are exactly the invertible matrices.