Adjoint group

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
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.