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 asubcategory 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 allmodules in $\mathcal{C}$ are pure injective. Then a homomorphism $f: A\to C$ ofleft $R$-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 $A^+$. Someapplications of this result are given.
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机译：假设$ R $为任意环，而$（-）^ + = \ Hom _ {\ mathbb {Z}}（-，\ mathbb {Q} / \ mathbb {Z}）$其中$ \ mathbb {Z} $为整数环和$ \ mathbb {Q} $是有理数的环，令$ \ mathcal {C} $是左$ R $ -modules的子类，而$ \ mathcal {D} $是right $的子类的子类R $模块，例如\ mathcal {C} $中的任何$ X \ $ \ mathcal {D} $中的$ X ^ + \和$ \ mathcal {C} $中的所有模块都是纯内射的。那么同构$ f：A \ to C $ of \ mathcal {C} $中带有$ C \的左$ R $ -modules是$ \ $ {$}的$ \ mathcal {C} $-（前）信封。 +：C ^ + \至A ^ + $是$ \ mathcal {D} $-$ A ^ + $的（前）封面。给出了该结果的一些应用。
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