# Adjoint group

## Definition

### For an arbitrary associative ring

Suppose is an associative ring (not necessarily unital). 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 .

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.:

### For a radical ring

`Further information: adjoint group of a radical ring`

We say that is a radical ring if the semigroup operation defined above makes all of a group. Equivalently, the adjoint group in this case is the whole of .

In the special case that is an algebra over a field, the adjoint group is termed an algebra group. The term is typically used in the context of finite fields and fields of positive characteristic.