Products of Nilpotents in a Quaternion Ring of Odd Order

Let R be a finite commutative local principal ring of order q n , where q = p r for an odd prime p and integer r with R/J(R) ≃ GF(q). We determine the number of elements in the quaternion ring H(R) that can be expressed as a product of at least 2n − 1 nilpotent elements, and show by example that this bound is sharp.