TITLE: Viewing Laurent polynomials as a localization of $R[X]$?
QUESTION [5 upvotes]: I believe that the Laurent polynomials over a ring $R$ are simply the localization of $R[X]$ at $S=\{X^n:n\geq 0\}$. 
However, I've always thought as Laurent polynomials as like "polynomials" in that they form a ring with specified addition and multiplication operations, not as equivalence classes in $S^{-1}R$. 
If you're coming at this from these two different viewpoints, what would it mean that the Laurent polynomials are the localization of $R[X]$ at $S$? Would that essentially mean that there is a ring isomorphism between then identifying an equivalence class with a Laurent polynomial as follows?
$$
(\sum_{i=0}^m r_iX^i)/X^n\leftrightarrow \sum_{i=0}^m r_iX^{i-n}
$$
Is that what is formally meant?

REPLY [5 votes]: I think you have the right idea. The two constructions are two approaches on forming the smallest extension of $R[X]$ in which $X$ is a unit:

Localize by the powers of $X$, because this is the smallest multiplicative subset containing $X$. So the desired extension is $R[X] \hookrightarrow R[X]_X$.
Add another variable to the ring and interpret it as the inverse of $X$, that is $R[X] \hookrightarrow R[X,Y]/(XY-1) \cong R[X,X^{-1}].$

Both constructions are indeed isomorphic by the map$$R[X,X^{-1}] \to  R[X]_X ,\; X \mapsto \frac{X}{1} , X^{-1}\mapsto \frac{1}{X}$$ which is induced by the corresponding map of $R[X,Y]$.