$C({\mathcal{X}}^{\ast })$-Cover and $C({\mathcal{X}}^{\ast })$-Envelope

Abstract

Let $R$ be any associative ring with unity and $\mathcal{X}$ be a class of $R$-modules of closed under direct sum (and summands) and with extension closed. We prove that every complex has an $C({\mathcal{X}}^{\ast })$-cover ($C({\mathcal{X}}^{\ast })$-envelope) if every module has an $\mathcal{X}$-cover ($\mathcal{X}$-envelope) where $C({\mathcal{X}}^{\ast })$ is the class of complexes of modules in $\mathcal{X}$ such that it is closed under direct and inverse limit.