Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$, $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements, $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements, and $\mathbb{C}(\mathbb{Z}^n)$ the field of fractions of $\mathbb{C}[\mathbb{Z}^n]$. We define a $\mathbb{C}$-linear map $\mathrm{I} : \mathcal{R} \rightarrow \mathbb{C}(\mathbb{Z}^n)$ as follows. If $f \in \mathcal{R}$ satisfies $Qf = P$ with $P, Q \in \mathbb{C}[\mathbb{Z}^n]$, we set $\mathrm{I}(f) = \frac{P}{Q}$. Show that $\mathrm{I}$ is well defined, and that it is a linear map with kernel $\mathcal{T}$ satisfying $\mathrm{I}(Pf) = P\,\mathrm{I}(f)$ for all $f \in \mathcal{R}$ and $P \in \mathbb{C}[\mathbb{Z}^n]$.
Let $n \geq 1$ be an integer. We denote by $\mathbb{C}[[\mathbb{Z}^n]]$ the $\mathbb{C}$-vector space of functions $f : \mathbb{Z}^n \rightarrow \mathbb{C}$, $\mathcal{R}$ the $\mathbb{C}$-vector space of rational elements, $\mathcal{T}$ the $\mathbb{C}$-vector space of torsion elements, and $\mathbb{C}(\mathbb{Z}^n)$ the field of fractions of $\mathbb{C}[\mathbb{Z}^n]$.
We define a $\mathbb{C}$-linear map $\mathrm{I} : \mathcal{R} \rightarrow \mathbb{C}(\mathbb{Z}^n)$ as follows. If $f \in \mathcal{R}$ satisfies $Qf = P$ with $P, Q \in \mathbb{C}[\mathbb{Z}^n]$, we set $\mathrm{I}(f) = \frac{P}{Q}$. Show that $\mathrm{I}$ is well defined, and that it is a linear map with kernel $\mathcal{T}$ satisfying $\mathrm{I}(Pf) = P\,\mathrm{I}(f)$ for all $f \in \mathcal{R}$ and $P \in \mathbb{C}[\mathbb{Z}^n]$.