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Submodule-closed subcategories of finite type

Volume 147 / 2017

Nguyen Viet Dung Colloquium Mathematicum 147 (2017), 125-144 MSC: 16G10, 16G60, 16D70, 16D90. DOI: 10.4064/cm6819-3-2016 Published online: 9 December 2016


Let $R$ be a left pure semisimple ring, and $\mathcal {C}$ a full subcategory of finitely generated left $R$-modules such that $\mathcal {C}$ is closed under finite direct sums and submodules. It is shown that if $\mathcal {C}$ has an infinite number of non-isomorphic indecomposable modules, then $\mathcal {C}$ contains a submodule-closed subcategory of finite type $\mathcal {A}$ (i.e. $\mathcal {A}$ has only finitely many non-isomorphic indecomposable modules) which is maximal among all submodule-closed subcategories of finite type in $\mathcal {C}$, and moreover $\mathcal {A}$ contains an indecomposable module which is not the source of a left almost split morphism in $R$-mod. If $R$ is an indecomposable hereditary left pure semisimple ring, a maximal submodule-closed subcategory of finite type of $R$-mod always contains the preprojective component of $R$-mod, and if such a ring $R$ has only two simple modules, the unique maximal submodule-closed subcategory of finite type in $R$-mod can be described explicitly.


  • Nguyen Viet DungDepartment of Mathematics
    Ohio University–Zanesville Zanesville, OH 43701, U.S.A.

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