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Recently Rim and Teply [11] found a necessary condition for the existence of -torsionfree covers with respect to a given hereditary torsion theory for the category -mod. This condition uses the class of -exact modules; i.e. the -torsionfree modules for which every its -torsionfree homomorphic image is -injective. In this note we shall show that the existence of -torsionfree covers implies the existence of -exact covers, and we shall investigate some sufficient conditions for the converse....
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.
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 of submodules of a torsionfree module consisting of all submodules of such that is torsionfree and such that every torsionfree homomorphic image of the relative injective hull of is relatively injective. The results obtained are applied to the study of relatively exact covers of torsionfree modules. As an application...
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 rational completion of an -module can be characterized as a -injective hull of with respect to a (hereditary) torsion functor depending on . Properties of a torsion functor depending on an -module are studied.
We give a new proof of the main result of [1] which does not use the classification of the finite simple groups.
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