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Relative exact covers

Ladislav Bican, Blas Torrecillas (2001)

Commentationes Mathematicae Universitatis Carolinae

Recently Rim and Teply [11] found a necessary condition for the existence of σ -torsionfree covers with respect to a given hereditary torsion theory for the category R -mod. This condition uses the class of σ -exact modules; i.e. the σ -torsionfree modules for which every its σ -torsionfree homomorphic image is σ -injective. In this note we shall show that the existence of σ -torsionfree covers implies the existence of σ -exact covers, and we shall investigate some sufficient conditions for the converse....

Relative hermitian Morita theory. Part II: Hermitian Morita contexts.

Pieter Verhaeghe, Alain Verschoren (1992)

Publicacions Matemàtiques

We introduce the notion of a relative hermitian Morita context between torsion triples and we show how these induce equivalences between suitable quotient categories of left and right modules.Due to the lack of involutive bimodules, the induced Morita equivalences are not necessarily hermitian, however.

Relatively exact modules

Ladislav Bican (2003)

Commentationes Mathematicae Universitatis Carolinae

Rim and Teply [10] investigated relatively exact modules in connection with the existence of torsionfree covers. In this note we shall study some properties of the lattice ℰ τ ( M ) of submodules of a torsionfree module M consisting of all submodules N of M such that M / N is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of M / N is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As an application...

Strongly graded left FTF rings.

José Gómez, Blas Torrecillas (1992)

Publicacions Matemàtiques

An associated ring R with identity is said to be a left FTF ring when the class of the submodules of flat left R-modules is closed under injective hulls and direct products. We prove (Theorem 3.5) that a strongly graded ring R by a locally finite group G is FTF if and only if Re is left FTF, where e is a neutral element of G. This provides new examples of left FTF rings. Some consequences of this Theorem are given.

The importance of rational extensions

Frans Loonstra (1988)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

The rational completion M ¯ of an R -module M can be characterized as a τ M -injective hull of M with respect to a (hereditary) torsion functor τ M depending on M . Properties of a torsion functor depending on an R -module M are studied.

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