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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
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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 (pre)envelopes, (pre)covers, duality, pure injective modules, character modules
MSC Classifications: 18G25, 16E30 show english descriptions Relative homological algebra, projective classes
Homological functors on modules (Tor, Ext, etc.)
18G25 - Relative homological algebra, projective classes
16E30 - Homological functors on modules (Tor, Ext, etc.)
 

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