Idempotent factorization of matrices over a Prüfer domain of rational functions

We consider the smallest subring D of R(X) containing every element of the form 1/(1 + x(2)), with x is an element of R(X). D is a Pruifer domain called the minimal Dress ring of R(X). In this paper, addressing a general open problem for Pruifer non Bezout domains, we investigate whether 2 x 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).