Abstract view
Duality of Preenvelopes and Pure Injective Modules


Published:20130722
Printed: Jun 2014
Zhaoyong Huang,
Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, P. R. China
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.