If $\mathbb{F}_{q^n}$ is a finite field with $q^n$ elements ($q$ being a power of a prime $p$) we have the trace map $tr^n_m:\mathbb{F}_{q^n}\rightarrow \mathbb{F}_{q^m}$ such that $x\mapsto x+F^m(x)+..+F^{n-m}(x)$, if $m\mid n$. So we can form the inverse limit of these maps, that will be $$ \left\{(x_n)_{n\in\mathbb{N}}\in \prod_{n\in\mathbb{N}} \mathbb{F}_{q^n}\;\middle|\;\text{$tr^n_m(x_n)=x_m$ (if $m|n$)}\right\} $$ (an abelian group and also an $\mathbb{F}_q$-module).

My questions are

"what" abelian group it is,

what is the relation with the algebraic closure of $\mathbb{F}_q$, and

can we put a ring structure on the inverse limit?