The Polypermutation Group of an Associative Ring

We study permutation polynomials through the device of the \emph{polypermutation group} of an associative ring $R$, denoted by ${\rm Pgr}(R)$. We derive some basic properties and compute the cardinality of ${\rm Pgr}(\mathbb{Z}/p^k)$ when $p\geq k$. We use this computation to determine the structure of ${\rm Pgr}(\mathbb{Z}/p^2)$.