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On q.f.d. modules and q.f.d. rings. (English) Zbl 1070.16002
Let \(R\) be an associative ring with identity, let \(M\) be a right \(R\)-module, and let \(\sigma[M]\) be the full subcategory of the category of all right \(R\)-modules consisting of all \(M\)-subgenerated modules. A module \(N\) is called ‘QFD’ (quotient finite dimensional) if every quotient module of \(N\) has finite Goldie dimension; \(N\) is called ‘locally QFD’ if every finitely generated submodule of \(N\) is QFD.
The aim of this paper is to give characterizations of locally QFD modules \(M\) in terms of when an arbitrary direct sum of injective (resp. weakly injective, tight, weakly tight) modules in \(\sigma[M]\) is tight, or weakly tight, or weakly injective. In particular, one retrieves a characterization of right QFD rings due to A. H. Al-Huzali, S. K. Jain and S. R. López-Permouth [J. Algebra 153, No. 1, 37-40 (1992; Zbl 0766.16007)].
MSC:
16D50 Injective modules, self-injective associative rings
16D60 Simple and semisimple modules, primitive rings and ideals in associative algebras
16D70 Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras)
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