Polyfunctions over Commutative Rings

A function $f:R\to R$, where $R$ is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative $p\in R[x]$. Based on this notion we introduce ring invariants which associate to $R$ the numbers $s(R)$ and $s(R';R)$, where $R'$ is the subring generated by $1$. For the ring $R=\mathbb Z/n\mathbb Z$ the invariant $s(R)$ coincides with the number theoretic \emph{Smarandache function} $s(n)$. If every function in a ring $R$ is a polyfunction, then $R$ is a finite field according to the R\'edei-Szele theorem, and it holds that $s(R)=|R|$. However, the condition $s(R)=|R|$ does not imply that every function $f:R\to R$ is a polyfunction. We classify all finite commutative rings $R$ with unit element which satisfy $s(R)=|R|$. For infinite rings $R$, we obtain a bound on the cardinality of the subring $R'$ and for $s(R';R)$ in terms of $s(R)$. In particular we show that $|R'|\leqslant s(R)!$. We also give two new proofs for the R\'edei-Szele theorem which are based on our results.