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Atomical Grothendieck categories. (English) Zbl 1073.18005
Let \(\mathcal{A}\) be a Grothendieck category and let \(Tors (\mathcal{A})\) be a lattice of localizing subcategories of \(\mathcal{A}\). The authors call a category atomical if \(Tors (\mathcal{A})\) consists of two elements. It is proved in the paper that if a Grothendieck category has a Gabriel dimension equal to \(\alpha\) than the lattice \(Tors(\mathcal{A})\) is semiartinian of Loewy length \(\alpha\).
The main result of the paper is a criterion for a Grothendieck category of being atomic. \(\mathcal{A}\) is atomical if and only if every nonzero injective object of \(\mathcal{A}\) is cogenerated. If \(\mathcal{A}\) is rich in projective objects then it is atomical if and only if every non-zero projective object of \(\mathcal{A}\) is a generator.
The results obtained are connected with well known results of Dlab about rings over which all torsions are trivial. Remark: Proposition 4.6 is true only for duo-rings. A noncommutative ring \(R\) of this kind is a ring of matrices over a local perfect ring.

18E15 Grothendieck categories (MSC2010)
16S90 Torsion theories; radicals on module categories (associative algebraic aspects)
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