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1. CMB 2013 (vol 57 pp. 318)

Huang, Zhaoyong
 Duality of Preenvelopes and Pure Injective Modules 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 modulesCategories:18G25, 16E30
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