Polynomial functions on finite commutative rings

Every function on a finite residue class ring D/I of a Dedekind domain D is induced by an integer-valued polynomial on D that preserves congruences mod I if and only if I is a power of a prime ideal. If Il is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all a, b is an element of R, if ab is an element of P-n then a is an element of P-k, b is an element of P-j with k + j greater than or equal to min(n, N), we determine the number of functions (as well as the number of permutations) on R arising from polynomials in R[x]. For a finite commutative local ring whose maximal ideal is of nilpotency 2, we also determine the structure of the semigroup of functions and of the group of permutations induced on R by polynomials in R[x].