Is there any standardized/formal way that the polynomial rings $$R[x_1,x_2,\ldots]$$ $$R[x_1,\ldots,x_n]$$ $$R[x]$$ are defined. Would it be proper to say that $\text{PolyRing}(I)$, the polynomial ring formed with variables indexed by $I$, is defined as the group ring
$$R[F^{ab}_I]$$ where the group is the free abelian group on generating set $I$.
Is there any problem with this definition? Thoughts?
(PS. why weren't "polynomial" rings of group rings over non-abelian free groups considered first) **random thought*
As noted in the comments, the construction is not that of a group ring, but that of a monoid ring, taken over the free abelian monoid on your set $I$. This construction is exactly the same as for the group ring construction: nowhere in that definition do you require inverses to make things work.
You might also want to think of polynomial rings as free objects themselves: indeed the polynomial ring with indeterminates indexed by $I$ and coefficients in $R$ is precisely the free commutative $R$-algebra on the set $I$.