Products of nilpotents in a quaternion ring of odd order

Abstract

Let $R$ be a finite commutative local principal ring of cardinality $q^n$, where $q = p^r$ for an odd prime $p$ and integer $r$ with $R/J(R) \simeq GF(q)$. 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.