It's a standard lemma that the gcd of the binomial coefficients $(k,n-k)$ (for $0\lt k\lt n$) is $p$ when $n=p^r$ for some $r>0$ (with $p$ prime) and $1$ in all other cases. It follows that for any $f(t)\in R[t]$ with properties as described, there is a finite splitting $R=\prod_{i=1}^mR_i$, where for each $i$ either
(a) $f(t)$ maps to $t$ in $R_i[t]$; or
(b) Some prime $p$ is zero in $R_i$, and $f(t)$ maps to $t^{p^r}$ in $R_i[t]$.
Thus, the general case is not very different from the ones you mentioned already. These ideas crop up in the study of formal group laws and the dual Steenrod algebra, so they are fairly well known among algebraic topologists.