Special precovering classes in comma categories

Abstract

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