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 $R$ .

Particular cases

Cases based on rings

Value of ring	Notion of group powered over that ring	Is the powering uniquely determined by the abstract group structure?	Does every group admit a powering over such a ring? Note that this condition would hold if the ring has a surjective ring homomorphism to $\mathbb {Z}$ .
$\mathbb {Z}$	usual concept of group	Yes	Yes
$\mathbb {Z} [\pi ^{-1}]$ where $\pi$ is a set of primes	group powered over a set of primes where the set of primes is $\pi$ . In other words, every element of the group has a unique $p^{th}$ root in the group.	Yes	No (unless $\pi$ is empty)
$\mathbb {Q}$	rationally powered group: unique $p^{th}$ roots for all primes $p$	Yes	No
$\mathbb {Z} /n\mathbb {Z}$ for a positive integer $n$	group whose exponent is finite and divides $n$	Yes	No
$\mathbb {Z} [t]$	group along with a set map from the group to itself (need not be a homomorphism) that commutes with powering and sends every element to an element that it commutes with.	No	Yes
$\mathbb {Q} [t]$	rationally powered group along with a set map from the group to itself that commutes with powering and sends every element to an element that commutes with it.	No	No

Unipotent algebraic groups

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