Why is $R((X))$ defined as follows? Let $R$ be a commutative ring.
Then $R((X))$ is defined as the set of all $\sum_{n\geq N} a_n X_n$ where $N\in\mathbb{Z}$ and is called "The Formal Laurent series". But why?
Why don't we consider the monoid ring $R[\mathbb{Z}]$ instead? I think $R[\mathbb{Z}]$ deserves to be called " the Formal laurent series" since it seems useful to interpret (Formal) Fourier series, and cannot understand why $R((X))$ deserves that name.
Moreover, is there a standard notation for the monoid ring $R[\mathbb{Z}]$?
 A: (Edit after a bit more research:) Another widespread notation for $R[\Bbb Z]$ is $R\Bbb Z$. The latter may be preferable if one wants to avoid confusion with the polynomial ring in either a single variable $\Bbb Z$ or the polynomial ring in countably many variables (that happen to be integers, but cannot be added or multiplied as such); then again, for some rings $R$, $R\Bbb Z$ might in some contexts be confused with the complex product $\{\,rz\mid r\in R, z\in\Bbb Z\,\}$, which is not the same (e.g., the complex product $\Bbb Z\Bbb Z$ is just $\Bbb Z$)
All of $R[X]$, $R[[X]]$, $R((X))$, $R[\Bbb Z]$ (and possibly more) have their own applications and hence their own justification. 
Note that $R[\Bbb Z]$ is (via the obvious embedding) in fact a small subring of $R((X))$, essentially it equals $\bigcup_{n\in\Bbb N}X^{-n}R[X]$ (just to emphasize that there are only finitely many non-zero coefficients) or simply $R[X,X^{-1}]\subsetneq R((X))$. 
On the other hand, we cannot use (what you may have had in mind actually) $R^\Bbb Z$, the set of all maps $\Bbb Z\to R$ (or if you want a series-like notation: $\sum_{n=-\infty}^\infty a_nX^n$) because for general $R$ this allows only addition, but not multiplication (at least not Cauchy-product-like). If $R$ carries a topology, we may be able to define the Cauchy product of two such series for some cases; we may even be able to pinpoint a suitable subgroup that happens to be closed under multiplication - but that is in no way a general approach.
