We consider the smallest subring D of R(X) containing every element of the form 1/(1+x²), with x ∈ R(X). D is a Prüfer domain called the minimal Dress ring of R(X). In this paper, addressing a general open problem for Prufer non B'ezout domains, we investigate whether 2×2 singular matrices over D can be decomposed as products of idempotent matrices. We show some conditions that guarantee the idempotent factorization in M2(D).