On the two definitions of degree of a function over an associative, commutative ring
Let R be an associative, commutative ring and let (: R<sup>m</sup> ^ R, where m ^ 0. Denote by deg<sub>n</sub> ( 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 deg<sub>RM</sub> ( denote the smallest integer n ^ -1 such that d<sub>v1</sub>... d<sub>Vn+1</sub> ( = = 0 for all v\,...,v<sub>n+</sub>\ £ R<sup>m</sup>. Here (d<sub>v</sub>ф)(х) = ф(х + v) - ф(х) for any v,x £ R<sup>m</sup> and any function ф: R<sup>m</sup> ^ R. If no such integer n exists, then we put deg<sub>n</sub> ( = то or deg<sub>RM</sub> ( = 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., deg<sub>n</sub> ( = deg<sub>RM</sub> ( for all m ^ 0 and all functions (: R<sup>m</sup> ^ 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.