The categorical treatment of fuzzy modules presents some problems, due to the well known fact that category is not abelian, and even normal. Our aim give a representation inside generalized modules, in fact, functor category, Mod−P, which Grothendieck category. To do that, first we consider preadditive P, defined by interval P=(0,1], build torsionfree class J hereditary torsion theory finally i...

Let R be ring and M a right R-module. This article introduces the concept of τ −⊕-supplemented modules as follows: Given a hereditary torsion theory in Mod-R with associated torsion functor τ we say that a module M is τ −⊕-supplemented when for every submodule N of M there exists a direct summand K of M such that M = N +K and N ∩K is τ−torsion, and M is called completely τ −⊕-supplemented if ev...

In an arbitrary Grothendieck category, we find necessary and sufficient conditions for the class of $\text{FP}_n$-injective objects to be a torsion class. By doing so, propose notion $n$-hereditary categories. We also define study $\text{FP}_n$-flat in categories with generating set small projective objects, provide several equivalent this torsion-free. end, present applications examples contex...

in this paper we introduce a generalization of m-small modules and discuss about the torsion theory cogenerated by this kind of modules in category . we will use the structure of the radical of a module in  and get some suitable results about this class of modules. also the relation between injective hull in  and this kind of modules will be investigated in this article.   for a module  we show...

Our first result provides a new characterization of Auslander algebras using property hereditary torsion pairs. The second shows an algebra $\Lambda$ is left or right glued if and only representation-finite. Finally, our third the module category any contains tilting with particular property, which we call property. Applications this are investigated.

Four cases of hereditary progressive dystonia with diurnal fluctuation were studied. All were sporadic; three of them mimicked spastic diplegia; and the fourth showed some similarity to torsion dystonia. Emotional or cognitive disturbance, or both, was seen in three. The correct diagnosis was suggested by fluctuating signs and symptoms, which worsened towards evening, but this was reached only ...

A torsion theory is called differential (higher differential) if a derivation (higher derivation) can be extended from any module to the module of quotients corresponding to the torsion theory. We study conditions equivalent to higher differentiability of a torsion theory. It is known that the Lambek, Goldie and any perfect torsion theories are differential. We show that these classes of torsio...