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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*

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    $\begingroup$ Note that the group ring of the integers is not isomorphic to the polynomial ring in one variable but to the ring of Laurent polynomials in one variable. Instead, the 'monoid ring' of the natural numbers would give the polynomial ring in one variable. $\endgroup$ Commented Mar 31, 2015 at 10:25
  • $\begingroup$ Just to be confusing, in some contexts, some authors call 'monoid rings' 'semigroup rings'. @MatthiasKlupsch $\endgroup$ Commented Mar 31, 2015 at 10:37

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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 $I$.

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