Polyfunctions over commutative rings

A function f:R→R, where R is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative p∈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=Z/nZ the invariant s(R) coincides with the number theoretic Smarandache or Kempner functions(n). If every function in a ring R is a polyfunction, then R is a finite field according to the Rédei–Szele theorem, and it holds that s(R)=|R|. However, the condition s(R)=|R| does not imply that every function f:R→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′|≤s(R)!. We also give two new proofs for the Rédei–Szele theorem which are based on our results.