Adjoint group of a radical ring

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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.