Covers and envelopes over Gorenstein rings.

*(English)*Zbl 0895.16001Let \(R\) 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).

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).

Reviewer: Y.Aoyama (Matsue / Shimane)

##### 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 |