Group powered over a unital ring

Definition
Suppose $$G$$ is a group and $$R$$ is a unital ring. A structure of $$G$$ as a group powered over $$R$$ includes an operation $$G \times R \to G$$ denoted by exponentiation, i.e., the output of $$(g,r)$$ is denoted $$g^r$$, satisfying the following conditions:


 * For $$r$$ an integer (modulo whatever is the characteristic of the ring), $$g^r$$ is the usual $$g^r$$.
 * For $$r,s \in R$$, $$(g^r)^s = g^{rs}$$.
 * For $$r,s \in R$$, $$(g^r)(g^s) = g^{r + s}$$

As a variety of algebras
For any fixed unital ring $$R$$, the groups powered over $$R$$ form a variety of algebras. This variety admits the variety of groups as a reduct, i.e., every group powered over a unital ring gives rise to a group.

Functors
Suppose $$R$$ and $$S$$ are unital rings and $$\varphi:R \to S$$ is a homomorphism of unital rings. Then, if $$G$$ is a group equipped with a powering structure over $$S$$, we naturally get a powering structure of $$G$$ over $$R$$.