  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.
