If $R$ is a ring of characteristic $p>0$, we define the Cartier Algebra $\displaystyle{\mathcal{C}=\bigoplus_{e\ge 0} \text{Hom}_R(R^{1/p^e},R)}$, where $R^{1/p^e}$ is the set of elements $r^{1/p^e}$, where $r\in R$, and is an $R$-module with left action $r\cdot s^{1/p^e}=(r^{p^e}s)^{1/p^e}$.
This is rather abstract for me, so I'm trying to compute $\mathcal{C}$ in the case that $R=\mathbb{Z}_p[x]$, and I would like confirmation that my calculation is correct (or incorrect). In this case $R^{1/q}=\mathbb{Z}[x^{1/q}]$ where $q=p^e$ for $e>0$. Then any $f$ in $\text{Hom}_R(R^{1/q},R)$ is defined by its values on $x^{1/q},x^{2/q},\cdots,x^{(q-1)/q}$, and we conclude that $\text{Hom}_R(R^{1/q},R)\simeq R^{\oplus q}$. Then $\mathcal{C}=\bigoplus_e R^{\bigoplus p^e}=R^\infty$.
I just want to know if this computation is correct. Searching up other sources give me papers and results on the Cartier algebra, but I haven't found many concrete examples.