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Covers and envelopes over Gorenstein rings. (English) Zbl 0895.16001
Let $$R$$ be a ring and $$\mathcal F$$ a category of left $$R$$-modules. An $$\mathcal F$$-cover of a left $$R$$-module $$M$$ is defined to be a linear map $$\phi\colon F\to M$$ with $$F\in{\mathcal F}$$ such that (a) for any linear map $$\psi\colon G\to M$$ with $$G\in{\mathcal F}$$, there is a linear map $$g\colon G\to F$$ such that $$\psi=\phi g$$, (b) every endomorphism $$f$$ of $$F$$ such that $$\phi f=\phi$$ is an automorphism. An $$\mathcal F$$-envelope of $$M$$ is defined dually. It is said that $$R$$ is a Gorenstein ring if it is left and right noetherian, $$\text{inj.dim} _RR<\infty$$ and $$\text{inj.dim} R_R<\infty$$. Assume $$R$$ is a Gorenstein ring, and let $$\mathcal L$$ be the class of left $$R$$-modules of finite projective dimension. A left $$R$$-module $$K$$ is called Gorenstein injective if $$\text{Ext}^1_R(L,K)=0$$ for any $$L\in \mathcal L$$, and we denote by $$\mathcal G$$ the class of Gorenstein injective left $$R$$-modules.
It is shown that if $$R$$ is a Gorenstein ring, then every left $$R$$-module has a $$\mathcal G$$-envelope and an $$\mathcal L$$-cover. The authors also consider the Gorenstein ring $$DG$$ with $$D$$ a discrete valuation ring and $$G$$ a finite group and remark that their results give a canonical way of lifting representations of G over $$\mathbb{Z}/(p)$$ to modular representations of $$G$$ over $$\widehat\mathbb{Z}_p$$ (the ring of $$p$$-adic integers).

##### MSC:
 16D50 Injective modules, self-injective associative rings 16D40 Free, projective, and flat modules and ideals in associative algebras 16P40 Noetherian rings and modules (associative rings and algebras) 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20C20 Modular representations and characters 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 16E10 Homological dimension in associative algebras
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