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group of units
Theorem.
The set of units of a ring forms a group with respect to ring multiplication.
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.
The group of the units of the ring is called the group of units of the ring. If is a field, is said to be the multiplicative group of the field.
Examples
1. When , then .
2. When , the ring of Gaussian integers, then .
3. When , then .
4. When where is a field, then .
5. When is the residue class ring modulo , then consists of the prime classes modulo , i.e. the residue classes satisfying .
Mathematics Subject Classification
16U60 Units, groups of units13A05 Divisibility; factorizations
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