Abstract We prove a correspondence between $\kappa$ -small fibrations in simplicial presheaf categories equipped with the injective or projective model structure (and left Bousfield localizations thereof) and relatively -compact maps their underlying quasi-categories for suitably large regular cardinals . thus obtain transition result weakly universal small (type-theoretic) Dugger–Rezk-style st...

The main goal of this paper is to introduce a new class in the category modules. It called quasi-invertibility monoform (briefly QI-monoform) This modules generalization Various properties and another characterization QI-monoform are investigated. So, we prove that an R-module M if only for each non-zero homomorphism f:M E(M), kernel not quasi-invertible submodule M. Moreover, cases under which...

If H is a nitely generated subgroup of the nitely generated group G, then H is undistorted in G for their word metrics ii the induced homomorphism on the zero-dimensionaì 1-homology group is injective. The Mayer-Vietoris exact sequence for`1-homology is established and applications are given to hyperbolic groups and relative hyperbolicity. The notion of relative distortion of two maps is introd...

An R-module is called semi-endosimple if it has no proper fully invariant essential submodules. For a quasi-projective retractable module MR we show that M is finitely generated semi-endosimple if and only if the endomorphism ring of M is a finite direct sum of simple rings. For an arbitrary module M , conditions equivalent to the semi-endosimplicity of its quasi-injective hull are found. As co...

We study the class of ADS rings and modules introduced by Fuchs [F]. We give some connections between this notion and classical notions such as injectivity and quasi-continuity. A simple ring R such that RR is ADS must be either right self-injective or indecomposable as a right Rmodule. Under certain conditions we can construct a unique ADS hull up to isomorphism. We introduce the concept of co...

In our recent work we gave a treatment of certain aspects of multiplication modules, projective modules, flat modules and cancellation-like modules via idealization. The purpose of this work is to continue our study and develop the tool of idealization, particularly in the context of closed, divisible injective, and simple modules. We determine when a ring R (M), the idealization of M , is a qu...

We will generalize the projective model structure in the category of unbounded complexes of modules over a commutative ring to the category of unbounded complexes of quasi-coherent sheaves over the projective line. Concretely we will define a locally projective model structure in the category of complexes of quasicoherent sheaves on the projective line. In this model structure the cofibrant obj...