Special precovering classes in comma categories

Abstract

Let T be a right exact functor from an abelian category ℬ into another abelian category \({\mathscr A}\). Then there exists a functor p from the product category \({\mathscr A} \times {\mathscr B}\) to the comma category (\(\left( {T \downarrow {\mathscr A}} \right)\)). In this paper, we study the property of the extension closure of some classes of objects in (\(\left( {T \downarrow {\mathscr A}} \right)\)), the exactness of the functor p and the detailed description of orthogonal classes of a given class \({\rm{P}}\left( {{\cal X},{\cal Y}} \right)\) in (\(\left( {T \downarrow {\mathscr A}} \right)\)). Moreover, we characterize when special precovering classes in abelian categories \({\mathscr A}\) and ℬ can induce special precovering classes in (\(\left( {T \downarrow {\mathscr A}} \right)\)). As an application, we prove that under suitable cases, the class of Gorenstein projective left Λ-modules over a triangular matrix ring \({\rm{\Lambda }} = \left( {\matrix{ R & M \cr 0 & S \cr } } \right)\) 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.