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Weak injective covers and dimension of modules. (English) Zbl 1363.18011

Let \(R\) be an associative ring with unit. A left \(R\)-module \(M\) is called supper finitely presented (s.f.p) if there exists an exact sequence \(\ldots\rightarrow P_{n} \rightarrow \ldots\rightarrow P_{1} \rightarrow P_{0} \rightarrow M \rightarrow 0\) and each left \(R\)-module \(P_{i}\) is finitely generated projective for all \(i\).
A left \(R\)-module \(M\) (resp. right \(R\)-module) is said to be weak injective (resp. weak flat) if \(\mathrm{Ext}^{1}_{R}(F, M) = 0\) (resp. \(\mathrm{Tor}_{1}^{R}(M, F) = 0\)) for any s.f.p left \(R\)-module \(F\).
In this paper the authors prove that every left \(R\)-module has a weak injective cover and a weak injective preenvelope. They give some criteria for computing the right and the left weak injective dimension of modules in terms of some properties of the left (resp. right) derived functors of the functor \(\mathrm{Hom}\) (resp. tensor product).

MSC:

18G25 Relative homological algebra, projective classes (category-theoretic aspects)
16E10 Homological dimension in associative algebras
16E30 Homological functors on modules (Tor, Ext, etc.) in associative algebras
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