Special precovering classes in comma categories

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 x B to the comma category (T down arrow A). In this paper, we study the property of the extension closure of some classes of objects in (T down arrow A), the exactness of the functor p and the detailed description of orthogonal classes of a given class p(X; Y) in (T down arrow A). Moreover, we characterize when special precovering classes in abelian categories A and B can induce special precovering classes in (T down arrow A). As an application, we prove that under suitable cases, the class of Gorenstein projective left Lambda-modules over a triangular matrix ring Lambda = ((R)(0) (M)(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.

Automated summary

This paper examines the properties of a right-exact functor from one abelian category to another, as well as the extension closure and exactness of certain classes of objects. It also explores how special precovering classes in different abelian categories can induce similar classes in the target comma category. Additionally, the paper provides examples where Gorenstein projective modules are special precovering in specific rings.