Given modules M and A, M is said to be A-RD-subinjective if for every RD-extension B of A, every \(f\in \textrm{Hom}(A, M)\) extends to \(\textrm{Hom}(B, M)\) . For a module M, the RD-subinjectivity domain of M is defined to be the collection of all modules A such that M is A-RD-subinjective. We investigate basic properties of RD-subinjectivity domains and provide characterizations for various types of rings and modules including p-injective modules, RD-coflat modules, von Neumann regular rings, RD-rings, Köthe rings, right Noetherian rings, and quasi-Frobenius rings in terms of RD-subinjectivity domains. Finally, we study the properties of RD-indigent modules and consider the structure of rings over which every (resp. simple) right module is RD-injective or RD-indigent.
