Injectivity and projectivity domains of a simple module over a commutative ring

Abstract

Let R be a commutative ring. We determine the cyclic modules in the projectivity domain and those in the injectivity domain of a simple R-module. An R-module M is called poor (p-poor) if for every R-module N, M is N-injective (M is N-projective) only if N is semisimple. We describe the structure of simple poor R-modules and simple p-poor R-modules. We show that every simple poor R-module is p-poor and the converse holds when R is noetherian or semilocal. It is shown that every simple R-module is poor if and only if every simple R-module is p-poor if and only if R is local or semisimple. We prove that every simple R-module is either p-poor or projective if and only if R is semisimple or R has exactly one essential maximal ideal. We conclude by studying the class of commutative rings R over which every simple R-module is either poor or injective.