## Definition

### For an arbitrary associative ring

Suppose $N$ is an associative ring (not necessarily unital). First, make $N$ a semigroup with the operation $x * y := x + y + xy$ (this is the multiplicative formal group law, but unlike other formal group laws, it works even without commutativity assumptions). The adjoint group of $N$ is the subgroup of this semigroup comprising the elements with two-sided inverses. The identity element for the adjoint group is $0 \in N$.

For convenience, and to avoid confusing the elements of $N$ with the same elements viewed as elements of the adjoint group, we denote the adjoint group element for $x \in N$ as the formal expression $1 + x$. The advantage of this is that the group multiplication now arises formally from the multiplication in $N$ and distributivity, i.e.:

$(1 + x)(1 + y) = 1 + x + y + xy = 1 + (x * y)$

We say that $N$ is a radical ring if the semigroup operation defined above makes all of $N$ a group. Equivalently, the adjoint group in this case is the whole of $1 + N$.
In the special case that $N$ 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.