Adjoint group

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Definition

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

Further information: adjoint group of 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.