group of units


The set E of units of a ring R forms a group with respect to ring multiplication.

Proof.  If u and v are two units, then there are the elements r and s of R such that  ru=ur=1  and  sv=vs=1.  Then we get that (sr)(uv)=s(r(uv))=s((ru)v)=s(1v)=sv=1,  similarly  (uv)(sr)=1.  Thus also uv is a unit, which means that E is closed under multiplication.  Because  1E  and along with u also its inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath r belongs to E, the set E is a group.

Corollary.  In a commutative ring, a ring product is a unit iff all are units.

The group E of the units of the ring R is called the group of units of the ring.  If R is a field, E is said to be the multiplicative groupMathworldPlanetmath of the field.


  1. 1.

    When  R=, then  E={1,-1}.

  2. 2.

    When  R=[i],  the ring of Gaussian integersMathworldPlanetmath, then  E={1,i,-1,-i}.

  3. 3.

    When  R=[3], then (  E={±(2+3)nn}.

  4. 4.

    When  R=K[X]  where K is a field, then  E=K{0}.

  5. 5.

    When  R={0+, 1+,,m-1+}  is the residue class ring modulo m, then  E consists of the prime classes modulo m, i.e. the residue classesMathworldPlanetmath l+ satisfying  gcd(l,m)=1.

Title group of units
Canonical name GroupOfUnits
Date of creation 2013-03-22 14:41:32
Last modified on 2013-03-22 14:41:32
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 24
Author pahio (2872)
Entry type Theorem
Classification msc 16U60
Classification msc 13A05
Synonym unit group
Related topic CommutativeRing
Related topic DivisibilityInRings
Related topic NonZeroDivisorsOfFiniteRing
Related topic PrimeResidueClass
Defines group of units of ring
Defines multiplicative group of field