Group of units

Definition

The group of units of a commutative unital ring is defined as the group whose underlying set is the set of elements possessing multiplicative inverses in the ring, and where the group operation is the multiplication operation of the ring.

Note that this group of units must be an Abelian group.

The group of units can also be defined for a noncommutative ring, in which case it will, in general, be a non-Abelian group.

Functoriality

The map from the category of commutative unital rings, to the category of Abelian groups, that sends a commutative unital ring to its group of units, is functorial.