The optimal version of Hua's fundamental theorem of geometry of square matrices - the low dimensional case

Let D be any division ring and p,q positive integers. The optimal version of Hua's fundamental theorem of geometry of square matrices has been known in all dimensions but the 2×2 case. We solve the remaining case by describing the general form of adjacency preserving maps ϕ:M2(D)→Mp×q(D). One of the main tools is a slight modification of known non-surjective versions of the fundamental theorem of affine geometry.