Products of idempotents in a quaternion ring

TL;DR

This paper characterizes elements in quaternion rings over finite local principal rings that can be expressed as products of idempotents, providing explicit formulas for their count and showing such elements are exactly those that are products of two idempotents.

Contribution

It establishes a precise criterion for products of idempotents in quaternion rings over finite local principal rings and derives explicit enumeration formulas.

Findings

An element in H(R) is a product of idempotents iff it is a product of two idempotents.

Explicit formulas are obtained for counting such elements.

The characterization simplifies understanding the structure of quaternion rings over these 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.