Classes of Preradicals Induced by Relative Injectivity

TL;DR

This paper introduces new classes of preradicals, such as prehereditary and weakly idempotent preradicals, to extend the concept of relative injectivity in module theory under weaker conditions.

Contribution

It defines and studies prehereditary, essentially idempotent, and weakly idempotent preradicals, broadening the framework for relative injectivity and torsion theories.

Findings

Extended classical injectivity results to broader contexts.

Introduced new classes of preradicals with weaker properties.

Facilitated analysis of relative injectivity under less restrictive assumptions.

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.