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 R is a finite commutative local ring with maximal ideal P of nilpotency N satisfying for all a, b∈R, if ab∈ P then a∈ P k , b∈ P j with k+ j ≥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]. Introduction Let R be a finite commutative ring with identity. Every polynomial f∈R[x] defines a function on R by substitution of the variable. Not every function φ:R → R is induced by a polynomial in R[x], however, unless R is a finite field. (Indeed, if the function with φ(0) = 0 and φ(r) = 1 for r ∈ R \ {0} is represented by f ∈ R[x], then f(x) = a1x + . . . + anx n and for every non-zero r ∈ R we have 1 = f(r) = (a1 + . . .+ anr )r, which shows r to be invertible.) This prompts the question how many functions on R are representable by polynomials in R[x]; and also, in the case that R=D/I is a residue class ring of a domain D with quotient field K , whether every function on R might be induced by a polynomial in K[x]? We will address these questions in sections 2 and 1, respectively. ‡ 1991 Math. Subj. Classification: Primary 13M10, 13B25; Secondary 11C08, 13F05, 11T06.