Unit (ring theory)

Not to be confused with Unit ring.

In mathematics, an invertible element or a unit in a (unital) ring $R$ is any element $u$ that has an inverse element in the multiplicative monoid of $R$ , i.e. an element $v$ such that

uv = vu = 1 R

, where

1 R

is the multiplicative identity.^[1]^[2]

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 zero 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.

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" or "identity", and say that $R$ is a "ring with unity" or a "ring with identity" rather than a "ring with a unit".

The multiplicative identity $1 R$ and its opposite $-1 R$ are always units. Hence, pairs of additive inverse elements^[3] $x$ and $- x$ are always associated.

Group of units

Main article: Green's relations § The H and D relations

The units of a ring $R$ form a group $U(R)$ under multiplication, the group of units of $R$ . Other common notations for $U(R)$ are $R *$ , $R \times$ , and $E(R)$ (from 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

r \sim 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 $f : R \to S$ induces a group homomorphism $U(f) : U(R) \to U(S)$ , 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 $U(R) = R ∖ {0}$ .

Examples

In the ring of integers $Z$ , the only units are $+1$ and $-1$ .
In the ring $Z / n Z$ of integers modulo $n$ , the units are the congruence classes $(mod n)$ represented by integers coprime to $n$ . They constitute the multiplicative group of integers modulo $n$ .
Any root of unity in a ring $R$ is a unit. (If $r n = 1$ , then $r n - 1$ is a multiplicative inverse of $r$ .)
If $R$ is the ring of integers in a number field, Dirichlet's unit theorem implies that the unit group of $R$ is a finitely generated abelian group. For example, we have $(\sqrt 5 + 2)(\sqrt 5 - 2) = 1$ in the ring $Z [1 + \sqrt 5 / 2]$ , and in fact the unit group of this ring is infinite. In general, the unit group of (the ring of integers of) a real quadratic field is infinite (of rank 1).
The unit group of the ring $M n (F)$ of $n \times n$ matrices over a field $F$ is the group $GL n (F)$ of invertible matrices.
The units of the real numbers $R$ are $R ∖ {0$ }.

References

↑ Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). John Wiley & Sons. ISBN 0-471-43334-9.
↑ Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.
↑ In a ring, the additive inverse of a non-zero element can equal to the element itself.

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