Adjoint group of a radical ring
First, make a semigroup with the operation (this is the multiplicative formal group law, but unlike other formal group laws, it works even without commutativity assumptions). The adjoint group of is the subgroup of this semigroup comprising the elements with two-sided inverses. The identity element for the adjoint group is .
1 plus notation
For convenience, and to avoid confusing the elements of with the same elements viewed as elements of the adjoint group, we denote the adjoint group element for as the formal expression . The advantage of this is that the group multiplication now arises formally from the multiplication in and distributivity, i.e.:
Interpretation inside unitization
The adjoint group can be thought of concretely as the coset for of in the unitization . Alternatively, if is an algebra over a commutative unital ring , the adjoint group can be thought of as the coset for of in the unitization .