Classical groups
Finite groups
Group schemes
Topological groups
Lie groups
Super-Lie groups
Higher groups
Cohomology and Extensions
Related concepts
For a ring, its group of units, denoted or , is the group whose elements are the elements of that are invertible under the product (units of the ring), and whose group operation is the multiplication in .
is an affine variety (in fact an affine algebraic group) over , namely .
This leads us to the following alternative perspective:
In a category with finite limits, with a ring object therein, the group of units of is the equalizer of the two maps , where is the ring multiplication and is the constant map with value the multiplicative identity.
The group of units of is equivalently the collection of morphisms from into the group of units
There is an adjunction
between the category of associative algebras over and that of groups, where forms the group algebra over and where assigns to an -algebra its group of units.
The multiplicative group of the ring of integers modulo is the multiplicative group of integers modulo n.
The group of units of the ring of adeles is the group of ideles. The topology on the idele group arises by considering as an affine variety in as above, and giving it the subspace topology. This is not the subspace topology induced by the inclusion into the ring of adeles.
The group of units of the -adic integers fits in an exact sequence
where the quotient is isomorphic to the cyclic group (see root of unity) and the kernel is, at least when , isomorphic to the additive group . Explicitly, for such the formal exponential map converges when and maps isomorphically onto the multiplicative group . The formal logarithm is also convergent for and provides the inverse.
By Hensel's lemma, the group of units has roots of unity and therefore the exact sequence above splits. This splitting descends to the quotient ring and its group of units, giving an isomorphism .
group of units/multiplicative group, Picard group, Brauer group
For instance:
Last revised on September 25, 2024 at 04:08:00. See the history of this page for a list of all contributions to it.