Adjoint group

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.:

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

For a radical ring

Further information: adjoint group of a radical ring

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.