Witt vector rings and quotients of monoid algebras

In a previous paper Cuntz and Deninger introduced the ring $C(R)$ for a perfect $\mathbb{F}_p$-algebra $R$. The ring $C(R)$ is canonically isomorphic to the $p$-typical Witt ring $W(R)$. In fact there exist canonical isomorphisms $\alpha_n \colon \mathbb{Z}R/I^n \xrightarrow{\sim} W_n(R)$. In this paper we give explicit descriptions of the isomorphisms $\alpha_n$ for $n\geq 2$ if $p\geq n$.