Permutation polynomials over finite rings

Let PPol(R) denote the group of permutation polynomial functions over the finite, commutative, unital ring R under composition. We generalize numerous results about permutation polynomials over Zpn to local rings by treating them under a unified manner. In particular, we provide a natural wreath product decomposition of permutation polynomial functions over the maximal ideal M and over the finite field R/M. We characterize the group of permutation polynomial functions over M whenever the condition M|R/M|={0} applies. Then we derive the size of PPol(R), thereby generalizing the same size formulas for Zpn. Finally, we completely characterize when the group PPol(R) is solvable, nilpotent, or abelian.