Năstăsescu, C.; Torrecillas, B. Atomical Grothendieck categories. (English) Zbl 1073.18005 Int. J. Math. Math. Sci. 2003, No. 71, 4501-4509 (2003). 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. Reviewer: Mykola Ya. Komarnytskyy (Lviv) Cited in 2 Documents MSC: 18E15 Grothendieck categories (MSC2010) 16S90 Torsion theories; radicals on module categories (associative algebraic aspects) Keywords:Grothendieck category; localizing subcategory; Gabriel dimension; semi-Artinian lattice; Loewy length; atomical Grothendieck category PDFBibTeX XMLCite \textit{C. Năstăsescu} and \textit{B. Torrecillas}, Int. J. Math. Math. Sci. 2003, No. 71, 4501--4509 (2003; Zbl 1073.18005) Full Text: DOI EuDML