Non-surjective Milnor patching diagrams

Milnor patching diagram is essentially the commutative square of rings, over which gluing of finitely generated projective modules is possible in the strongest sense. Necessary and sufficient conditions for a square to be Milnor patching diagram were studied by Milnor, Beauville-Laszlo and Landsburg. We relate this question to determinant-induced factorization in matrix rings to construct a series of non-surjective Milnor patching diagrams, settling the question of Landsburg, and make a step towards the classification of such examples. Also we consider a possible generalization of the notion of Milnor patching diagram to arbitrary subcategories of modules and obtain a classification result for this setting.