Products of idempotents in a quaternion ring

TL;DR

This work characterizes products of idempotents in quaternion rings over finite local principal rings. It shows that every product of idempotents in $H(R)$ is a product of two idempotents, leveraging an isomorphism $H(R) \cong M_2(R)$ when $2$ is invertible and an orbit-based classification of matrices $M(a,b)$. It derives an explicit counting formula for the number of elements in $H(R)$ that admit such factorizations, distinguishing the cases of invertible and non-invertible $2$; the formula in the invertible case is $|Z|=\frac{q^2+1+q^{3n}(q+1)^2}{q^2+q+1}$. Overall, the paper advances both structural and enumerative understanding of idempotent factorization in noncommutative quaternion rings over finite local rings.

Abstract

Let $R$ be a finite commutative local principal ring, and let $H(R)$ denote the corresponding quaternion ring. We show that an element of $H(R)$ is a product of idempotents if and only if it can be expressed as a product of two idempotents. Moreover, we obtain an explicit formula for the number of elements of $H(R)$ admitting such a factorization.

Theorems & Definitions (24)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• Theorem 2.4
• proof
• Lemma 3.1
• proof
• Lemma 3.2