On minimal absolutely pure domain of RD-fllat modules

Given modules A(R) and B-R, B-R is called absolutely A(R)-pure if for every extension C-R of B-R, A circle times B -> A circle times C is a monomorphism. The class (Fl) under bar (-1)(A(R)) ={B-R : B-R is absolutely A(R)-pure} is called the absolutely pure domain of a module A(R). If B-R is divisible, then all short exact sequences starting with B is RD-pure, whence B is absolutey A-pure for every RD-flat module A(R). Thus the class of divisible modules is the smallest possible absolutely pure domain of an RD-flat module. In this paper, we consider RD-flat modules whose absolutely pure domains contain only divisible modules, and we referred to these RD-flat modules as rd-indigent. Properties of absolutely pure domains of RD-flat modules and of rd-indigent modules are studied. We prove that every ring has an rd-indigent module, and characterize rd-indigent abelian groups. Furthermore, over (commutative) SRDP rings, we give some characterizations of the rings whose nonprojective simple modules are rd-indigent.