in: D.E. Dobbs, M. Fontana, S.-E. Kabbaj (eds.),
Advances in Commutative Ring Theory (Proc. of 1997 Fez Conf)
Lecture Notes in Pure and Appl. Mathematics 205, Dekker 1999,
pp 323-336.

Polynomial functions on finite commutative rings

Abstract: We answer, for Dedekind domains, a question of W. Narkiewicz: If I is an ideal of finite index of a Dedekind doman D, then every function on D/I 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.

We also summarize and generalize the known results on the number of functions and the number of permutations on a finite ring induced by polynomials with coefficients in the ring itself; and 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].

1991 Mathematics Subject Classification:
Primary 13M10, 13B25; Secondary: 11C08, 13F05, 11T06.

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