## Definition

Suppose $N$ is an radical ring, i.e., an associative ring where, for every $x \in N$, there exists $y \in N$ such that $x + y + xy = 0$. Note that $N$ cannot be a unital ring, because it's not possible to find a $y$ that works for $-1$.

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 $0 \in N$.

### 1 plus notation

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.:

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

### Interpretation inside unitization

The adjoint group can be thought of concretely as the coset $1 + N$ for $N$ of $1$ in the unitization $N + \mathbb{Z}$. Alternatively, if $N$ is an algebra over a commutative unital ring $R$, the adjoint group can be thought of as the coset $1 + N$ for $N$ of $1$ in the unitization $1 + N$.