h-Fold Sums from a Set with Few Products

Before we state our main theorems, we begin with some notation: given a finite subset A of some commutative ring, we let A + A denote the set of sums a + b, where a, b ∈ A; and, we let A.A denote the set of products ab, a, b ∈ A. When three or more sums or products are used, we let kA denote the k-fold sumset A+A+ · · ·+A, and let A denote the k-fold product set A.A...A. Lastly, by d ∗ A we mean the set A dilated by the scalar d, which is just the set da, a ∈ A.