[Part II, cf. Kyungpook Math. J. 21, 171-176 (1981; Zbl 0516.16007).] A left A-module M is called FK-injective if, for any left submodule F isomorphic to a complement nonzero left submodule K of M, any relative complement C of \({}_ AF\) in \({}_ AM\) and any left submodule L of M containing \(C+F\), every epimorphism of \({}_ AL\) into \({}_ AK\) extends to an endomorphism of \({}_ AM\). The ring A is called left FK-injective iff \({}_ AA\) is FK-injective. In this paper, FK-injective is introduced to study von Neumann regular rings and continuous rings. It is shown that a ring A is left continuous regular iff A is a left FK-injective ring satisfying any of the following: a) A is left nonsingular; b) A is fully left idempotent; c) every simple left A-module is either p-injective or projective. Also conditions for certain nonsingular modules to be completely reducible and injective are given.