# Adjoint group of a radical ring

## Definition

Suppose is an radical ring, i.e., an associative ring where, for every , there exists such that . Note that cannot be a unital ring, because it's not possible to find a that works for .

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 .