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Gorenstein injective and projective modules. (English) Zbl 0845.16005
Let \(\mathcal F\) denote some class of modules. The first author previously defined \(\mathcal F\)-precovers, \(\mathcal F\)-preenvelopes, \(\mathcal F\)-covers and \(\mathcal F\)-envelopes [Isr. J. Math. 39, 189-209 (1981; Zbl 0464.16019)]. If \(\mathcal E\) is the class of injective left \(R\)-modules; then every left \(R\)-module admits an \(\mathcal E\)-cover. This fact is used to define left derived functors of Hom, which are utilized to characterize and prove properties of Gorenstein injective modules. For left noetherian rings, results on reconstruction of minimal injective resolutions are presented and for Gorenstein rings, minimal injective resolutions and resolvents are used to generate Gorenstein injective modules. The paper also contains results on finitely generated Gorenstein projective modules and a new way is shown to generate indecomposable Gorenstein projective modules from other such modules. The paper is concluded by a section on the existence of Gorenstein injective preenvelopes of modules.

MSC:
16D50 Injective modules, self-injective associative rings
16D40 Free, projective, and flat modules and ideals in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
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