In this paper we prove that the extension of pure injective module is pure injective if and only if the cotorsion envelope and the pure injective envelope of any $R$-module $M$ are isomorphic over $M$. And we prove that if the product of pure injective envelopes of flat modules is a pure injective envelope and the product of flat covers is a flat cover, then the product of cotorsion envelopes is a cotorsion envelope.
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机译：在本文中，我们证明，仅当任何$ R $-模块$ M $的扭曲包络和纯内射包络在$ M $上同构时，纯内射模的扩展才是纯内射。并且我们证明，如果平面模块的纯内射包络的产品是纯内射包络，而平盖的产物是平盖，则扭曲包络的乘积就是扭曲包络。
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