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Under semi-weak and weak compatibility conditions of bimodules, we establish necessary and sufficient conditions of Gorenstein-projective modules over rings of Morita contexts with one bimodule homomorphism zero. This extends greatly the results on triangular matrix Artin algebras and on Artin algebras of Morita contexts with two bimodule homomorphisms zero in the literature, where only sufficient conditions are given under a strong assumption of compatibility of bimodules. An application is provided to describe Gorenstein-projective modules over noncommutative tensor products arising from Morita contexts. Our results are proved under a general setting of noetherian rings and modules instead of Artin algebras and modules. [ABSTRACT FROM AUTHOR]

Let T be a right exact functor from an abelian category ℬ into another abelian category A . Then there exists a functor p from the product category A × ℬ to the comma category ( (T ↓ A) ). In this paper, we study the property of the extension closure of some classes of objects in ( (T ↓ A) ), the exactness of the functor p and the detailed description of orthogonal classes of a given class P (X , Y) in ( (T ↓ A) ). Moreover, we characterize when special precovering classes in abelian categories A and ℬ can induce special precovering classes in ( (T ↓ A) ). As an application, we prove that under suitable cases, the class of Gorenstein projective left Λ-modules over a triangular matrix ring Λ = ( R M 0 S ) is special precovering if and only if both the classes of Gorenstein projective left R-modules and left S-modules are special precovering. Consequently, we produce a large variety of examples of rings such that the class of Gorenstein projective modules is special precovering over them. [ABSTRACT FROM AUTHOR]

An additive functor $$F:\mathcal{A} \to \mathcal{B}$$ between additive categories is said to be objective, provided any morphism f in $$\mathcal{A}$$ with F( f) = 0 factors through an object K with F( K) = 0. We concentrate on triangle functors between triangulated categories. The first aim of this paper is to characterize objective triangle functors F in several ways. Second, we are interested in the corresponding Verdier quotient functors $$V_F :\mathcal{A} \to \mathcal{A}/KerF$$, in particular we want to know under what conditions V is full. The third question to be considered concerns the possibility to factorize a given triangle functor F = F F with F a full and dense triangle functor and F a faithful triangle functor. It turns out that the behavior of splitting monomorphisms and splitting epimorphisms plays a decisive role. [ABSTRACT FROM AUTHOR]