  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 $M_2(D)$.
