Idempotent factorization of matrices over a Pr\"ufer domain of rational functions

TL;DR

This paper investigates whether 2x2 singular matrices over a specific Pr"ufer domain of rational functions can be decomposed into products of idempotent matrices, addressing an open problem in algebra.

Contribution

It provides conditions under which singular matrices over the minimal Dress ring of rational functions can be factored into idempotent matrices, advancing understanding of matrix factorizations over Pr"ufer domains.

Findings

Identifies conditions for idempotent factorization in M_2(D)

Addresses an open problem for Pr"ufer non-Bézout domains

Contributes to algebraic matrix theory over specialized rings

Abstract

We consider the smallest subring $D$ of $\mathbb{R}(X)$ containing every element of the form $1/(1+x^2)$, with $x\in \mathbb{R}(X)$. $D$ is a Pr\"ufer domain called the minimal Dress ring of $\mathbb{R}(X)$. In this paper, addressing a general open problem for Pr\"ufer non B\'ezout domains, we investigate whether $2\times 2$ singular matrices over $D$ can be decomposed as products of idempotent matrices. We show some conditions that guarantee the idempotent factorization in $M_2(D)$.