On the two definitions of degree of a function over an associative, commutative ring
Let R be an associative, commutative ring and let (: Rm ^ R, where m ^ 0. Denote by degn ( the smallest integer n ^ -1 such that ( can be represented by an m-variate polynomial of degree n over R. (By convention, the degree of the zero polynomial is -1.) Also, let degRM ( denote the smallest integer n ^ -1 such that dv1... dVn+1 ( = = 0 for all v\,...,vn+\ £ Rm. Here (dvф)(х) = ф(х + v) - ф(х) for any v,x £ Rm and any function ф: Rm ^ R. If no such integer n exists, then we put degn ( = то or degRM ( = то, respectively. In this paper, we study the problem of characterizing the class D of all associative, commutative rings R such that these degrees coincide for functions over R, i.e., degn ( = degRM ( for all m ^ 0 and all functions (: Rm ^ R. We solve this problem when the additive group R of the ring R belongs to some large classes of abelian groups. Namely, our main results are as follows: 1) if R is torsion or finitely generated, then R £ D if and only if R = Z/dZ for some square-free integer d ^ 1; 2) if R is not reduced, then R £ D if and only if R = (Z/dZ) ф Q for some square-free integer d ^ 1; 3) if R is a direct sum of rank 1 subgroups, then R £ D if and only if R = Z/dZ or R = (Z/dZ) ф Q for some square-free integer d ^ 1; 4) if R is reduced and cotorsion, then R £ D if and only if R = П (Z/pZ) for some set P of peP prime numbers. The proof of these results is based on the fact that any ring in D is an E-ring.