Classes of Preradicals Induced by Relative Injectivity

Abstract

This paper analyzes new classes of preradicals defined as weak forms of left exact and idempotent preradicals. We introduce prehereditary preradicals to generalize the hereditary property, and essentially idempotent and weakly idempotent preradicals to generalize idempotency. Our motivation is to study the concept of relative injectivity. We show that these new classes facilitate the extension of classical results on injectivity with respect to a hereditary torsion theory to a broader context, requiring weaker assumptions.