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Zbl 0716.16004
Tonolo, Alberto
On a class of strongly quasi injective modules. (English)
[J] Rend. Semin. Mat. Univ. Padova 82, 115-131 (1989). ISSN 0041-8994

Let ${}\sb RK$ be a left unitary module over a ring R with unity and ${\cal C}(\sb RK)$ be the category of topological left R-modules, isomorphic to closed submodules of powers of ${}\sb RK$ endowed with the discrete topology. For $A=End(\sb RK)$ consider the category Mod-A of right unitary A-modules and its subcategory ${\cal D}(K\sb A)$ cogenerated by $K\sb A$. The bimodules ${}\sb RK\sb A$ which generate (in the standard way) a duality between ${\cal C}(\sb RK)$ and ${\cal D}(K\sb A)$ such that the K-valued continuous characters in ${\cal C}(\sb RK)$ can be extended where characterized by {\it C. Menini} and {\it A. Orsatti} [Ann. Mat. Pura Appl., IV Ser. 127, 187-230 (1981; Zbl 0476.16029)]. The aim of the paper under review is to describe the particular case when ${\cal D}(K\sb A)=Mod$-A, i.e. $K\sb A$ is a cogenerator of Mod-A. The author shows that this occurs precisely when ${\cal C}(\sb RK)$ is an abelian category. Using results of {\it A. Orsatti} and the reviewer [Rend. Accad. Naz. Sci. Detta XL, V. Ser., Mem. Mat. 8, 143-183 (1984; Zbl 0562.16023)] he obtains various constraints on the structure of ${}\sb RK$. A detailed characterization is given in the more particular case when $K\sb A$ is an injective cogenerator of Mod-A. Finally, an example of a duality of the above type such that $K\sb A$ is a non-injective cogenerator of Mod-A is provided.
[D.Dikranjan]

MSC 2000:: *16D90 Module categories (assoc. rings and algebras)
16D50 Injective modules, self-injective rings (assoc. rings and algebras)
16W80 Topological and ordered associative rings and modules

Keywords: good duality; linear compactness; category of topological left R-modules; right unitary A-modules; bimodules; abelian category; injective cogenerator

Citations: Zbl 0476.16029; Zbl 0562.16023

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