Modules and abelian groups with a restricted domain of projectivity

In this paper, we study the modules whose projectivity domains are contained in the class of all pure-split modules and call them p-impecunious modules. Every quotient of a p-impecunious R-module is p-impecunious if and only if R is right pure semisimple. An abelian group A is p-impecunious if all p-components of A are nonzero and B-p(A)not equal 0 for some prime p where B-p(A) is the basic subgroup of the p-component T-p(A). The converse is true if A is torsion. We characterize Dedekind domains over which there are injective p-impecunious modules.