Converting a polynomial ring to a numerical ring (transport of structure)

If $\,R\,$ is a ring (or any algebraic structure) then one can transport the structure of $\,R\,$ to any set $\,S\,$ of the same cardinality, by push/pulling the algebra operations along any bijection of their underlying sets. When $\,S\,$ = $\,\Bbb N\,$ or $\,\Bbb Z,\,$ this can be viewed simply as indexing (or coding) the elements of $\,R\,$ (e.g. in computer representations of $\,R\,$ where indices are memory addresses).

For example, to add $\,m,n\in\Bbb Z\,$ we first unindex them to $\,i^{-1}(m),\,i^{-1}(n)\in R,\,$ then perform the addition in $\,R,\,$ then index the result, i.e. the transported addition $\,\oplus\,$ in $\,S=\Bbb Z\,$ is

$$ m \oplus n\, =\, i\,(i^{-1}(m)+i^{-1}(n))$$

and analogously for all other operations of $\,R.\,$ This implies that the unindex map $\,i^{-1}$ is a ring homomorphism, i.e. $\, i^{-1}(m\oplus n) = i^{-1}(m)+i^{-1}(n),\,$ and similarly for other operations, yielding a ring isomorphism $\, R\cong \Bbb Z\,$ with said transported operations.