For a ring, its group of units, denoted or , is the group whose elements are the elements of that are invertible under the product, and whose group operation is the multiplication in .
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 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 .