Further Notes on Tightness

TL;DR

This paper explores the conditions under which tight and essentially tight modules coincide, examining the impact of essentiality restrictions on embeddings and introducing a new class of modules that generalize existing notions.

Contribution

It investigates when tightness and essential tightness are equivalent and introduces a new class of modules based on partial essentiality of embeddings.

Findings

Identifies conditions where tight and essentially tight modules are the same

Defines a new class of modules with partial essentiality constraints

Provides insights into the structure of modules with restricted embeddings

Abstract

Tight and essentially tight modules generalize weakly injective modules. Essential tightness requires embeddings to be essential. This restriction makes the two notions totally different. In this note, we investigate cases when those two notions are the same. Moreover, we look at the cases when essentiallity is imposed only on one of the embeddings rather than both. This allows defining a special class of tight and essentially tight modules and a generalization of both.