Local polynomial functions on factor rings of the integers

Abstract

AbstractLet A be a universal algebra. A function ϕ Ak-A is called a t-local polynomial function, if ϕ can ve interpolated on any t places of Ak by a polynomial function— for the definition of a polynomial function on A, see Lausch and Nöbauer (1973), Let Pk(A) be the set of the polynomial functions, LkPk(A) the set of all t-local polynmial functions on A and LPk(A) the intersection of all LtPk(A), then . If A is an abelian group, then this chain has at most five distinct members— see Hule and Nöbauer (1977)— and if A is a lattice, then it has at most three distinct members— see Dorninger and Nöbauer (1978). In this paper we show that in the case of commutative rings with identity there does not exist such a bound on the length of the chain and that, in this case, there exist chains of even infinite length.