Duality of Preenvelopes and Pure Injective Modules

TL;DR

This paper explores the duality between preenvelopes and pure injective modules over arbitrary rings, establishing conditions under which homomorphisms serve as (pre)envelopes based on their duals, with various applications.

Contribution

It introduces a duality framework linking preenvelopes and pure injective modules via module duality, extending understanding in module theory.

Findings

Characterizes when a homomorphism is a preenvelope via its dual cover.

Establishes duality conditions for pure injective modules and preenvelopes.

Provides applications demonstrating the theoretical results.

Abstract

Let $R$ be an arbitrary ring and $(-)^+=\Hom_{\mathbb{Z}}(-, \mathbb{Q}/\mathbb{Z})$ where $\mathbb{Z}$ is the ring of integers and $\mathbb{Q}$ is the ring of rational numbers, and let $\mathcal{C}$ be a subcategory of left $R$-modules and $\mathcal{D}$ a subcategory of right $R$-modules such that $X^+\in \mathcal{D}$ for any $X\in \mathcal{C}$ and all modules in $\mathcal{C}$ are pure injective. Then a homomorphism $f: A\to C$ of left $R$-modules with $C\in \mathcal{C}$ is a $\mathcal{C}$-(pre)envelope of $A$ provided $f^+: C^+\to A^+$ is a $\mathcal{D}$-(pre)cover of $A^+$. Some applications of this result are given.