location:  Publications → journals → CMB
Abstract view

Duality of Preenvelopes and Pure Injective Modules

Published:2013-07-22
Printed: Jun 2014
• Zhaoyong Huang,
Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P. R. China
 Format: LaTeX MathJax PDF

Abstract

Let $R$ be an arbitrary ring and $(-)^+=\operatorname{Hom}_{\mathbb{Z}}(-, \mathbb{Q}/\mathbb{Z})$ 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.
 Keywords: (pre)envelopes, (pre)covers, duality, pure injective modules, character modules
 MSC Classifications: 18G25 - Relative homological algebra, projective classes 16E30 - Homological functors on modules (Tor, Ext, etc.)