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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.

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