Euclidean pairs and quasi-Euclidean rings

We study the interplay between the classes of right quasi-Euclidean rings and right K-Hermite rings, and relate them to projective-free rings and Cohn's GE2-rings using the method of noncommutative Euclidean divisions and matrix factorizations into idempotents. Right quasi-Euclidean rings are closed under matrix extensions, and a left quasi-Euclidean ring is right quasi-Euclidean if and only if it is right Bézout. Singular matrices over left and right quasi-Euclidean domains are shown to be products of idempotent matrices, generalizing an earlier result of Laffey for singular matrices over commutative Euclidean domains.