Idempotent factorizations of singular 2 × 2 matrices over quadratic integer rings

Let D be the ring of integers of a quadratic number field Q[ √ d]. We study the factorizations of 2 × 2 matrices over D into idempotent factors. When d < 0 there exist singular matrices that do not admit idempotent factorizations, due to results by Cohn and by the authors Cozzu and Zanardo. We mainly investigate the case d > 0. We employ Vaseršte˘ın’s result that SL2(D) is generated by elementary matrices, to prove that any 2 × 2 matrix with either a null row or a null column is a product of idempotents. As a consequence, every column-row matrix admits idempotent factorizations.