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![[parent]](http://images.planetmath.org:8080/images/uparrow.png)
group of units
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(Theorem)
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Proof. If  and  are two units, then there are the elements  and  of  such that
 and
 . Then we get that
 , similarly
 . Thus also  is a unit, which means that  is closed under multiplication. Because  and along with  also its inverse  belongs to  , the set  is a group.
Corollary. In a commutative ring, a ring product is a unit iff all factors are units.
Examples
- When
 , then
 .
- When
![$ R = \mathbb{Z}[i]$](http://images.planetmath.org:8080/cache/objects/6301/l2h/img21.png "$ R = \mathbb{Z}[i]$") , the ring of Gaussian integers, then
 .
- When
![$ R = \mathbb{Z}[\sqrt{3}]$](http://images.planetmath.org:8080/cache/objects/6301/l2h/img23.png "$ R = \mathbb{Z}[\sqrt{3}]$") , then
 .
- When
![$ R = K[X]$](http://images.planetmath.org:8080/cache/objects/6301/l2h/img25.png "$ R = K[X]$") where  is a field, then
 .
- When
 is the residue class ring modulo  , then  consists of the prime classes modulo  , i.e. the residue classes
 satisfying
 .
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"group of units" is owned by pahio.
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(view preamble)
Cross-references: residue classes, prime classes, residue class ring, field, Gaussian integers, iff, ring product, commutative ring, inverse, multiplication, closed under, ring multiplication, group, ring, units
There are 20 references to this entry.
This is version 20 of group of units, born on 2004-10-05, modified 2008-03-30.
Object id is 6301, canonical name is GroupOfUnits.
Accessed 4116 times total.
Classification:
| AMS MSC: | 13A05 (Commutative rings and algebras :: General commutative ring theory :: Divisibility) | | | 16U60 (Associative rings and algebras :: Conditions on elements :: Units, groups of units) |
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Pending Errata and Addenda
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