Let ℱ𝒫∞R be the class of all left R-modules M which has a projective resolution by finitely generated projectives. An exact sequence 0 → A → B → C → 0 of right R-modules is called neat if the sequence 0 → A ⊗RG → B ⊗RG → C ⊗RG → 0 is exact for any G ∈ℱ𝒫∞R. An exact sequence 0 → M → N → L → 0 of left R-modules is called clean if the sequence 0 →HomR(G,M) →HomR(G,N) →HomR(G,L) → 0 is exact for any G ∈ℱ𝒫∞R. We prove that every R-module has a clean-projective precover and a neat-injective envelope. A morphism f of right R-modules is called a neat-phantom morphism if Tor1R(f,G) = 0 for any G ∈ℱ𝒫∞R. A morphism g of left R-modules is said to be a clean-cophantom morphism if ExtR1(G,g) = 0 for any G ∈ℱ𝒫∞R. We establish the relationship between neat-phantom (respectively, clean-cophantom) morphisms and neat (respectively, clean) exact sequences. Also, we prove that every R-module has a neat-phantom cover with kernel neat-injective and a clean-cophantom preenvelope with cokernel clean-projective.

中文翻译：

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Neat-phantom 和 clean-cophantom 态射

让ℱ𝒫∞R成为所有剩下的人R-模块米它通过有限生成的投影具有投影分辨率。准确的顺序0 → 一种 → 乙 → C → 0权利的R-modules 如果序列被称为整洁0 → 一种 ⊗RG → 乙 ⊗RG → C ⊗RG → 0对任何一个都是准确的G ∈ℱ𝒫∞R. 准确的顺序0 → 米 → ñ → 大号 → 0左边的R-modules 被称为 clean 如果序列0 →霍姆R(G,米) →霍姆R(G,ñ) →霍姆R(G,大号) → 0对任何一个都是准确的G ∈ℱ𝒫∞R. 我们证明每R-module 有一个干净的投影前盖和一个整洁的内射信封。态射F权利的R-modules 称为净幻态射，如果托尔1R(F,G) = 0对于任何G ∈ℱ𝒫∞R. 态射G左边的R-modules 被认为是一个干净的同态态射，如果分机R1(G,G) = 0对于任何G ∈ℱ𝒫∞R. 我们建立了整洁-幻象（分别是，干净-共幻）态射和整洁（分别是，干净）精确序列之间的关系。此外，我们证明每个R-module 有一个带有内核整洁内射的整洁幻影覆盖和一个带有 cokernel clean-projective 的 clean-cophantom preenvelope。