Adjoint group of a radical ring

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 $1+N$.