Products of nilpotents in a quaternion ring of odd order

TL;DR

This paper investigates the structure of quaternion rings over finite local principal rings of odd order, specifically focusing on elements that can be expressed as products of nilpotent elements, establishing bounds and sharpness of these bounds.

Contribution

It determines the number of elements in quaternion rings over certain finite rings that are products of multiple nilpotent elements, providing sharp bounds and examples.

Findings

Established the exact count of such elements for given ring parameters

Proved the bound of 2n-1 nilpotent factors is sharp

Provided explicit examples illustrating the bounds

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.