On weak injectivity of direct sums of modules

Abstract

Generalizing a notion defined by Jain and L6pez-permouth, we call a module Q e olMl weakly injective (resp. weakly tight) inolMlif, for every finitely generated submodule Nofthe M-injectlehu[ f,Niscontainedinasubmodule y otQ suchthatr - p (resp. Nisfinitely O-cogenerated). For some classes M of weakly injectives in o [M], we study the inJtances in which direct sums of modules fromM are weakly injective in ofMl. ln particular, we getnecessary and sufficient conditions for f -weak injectivity or !-weak tightness of the injective hull of a simple module. As a consequence, we get chancteizations for q.f.d. rings by mians of weakly injective modules given by Al-Huzali, Jain, and L6pez-permouth.