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 $$1 + N$$.