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تسجيل مستخدم جديدSpecial precovering classes in comma categories
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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 $mathscr{A}timesmathscr{B}$ to the comma category $(Tdownarrowmathscr{A})$. In this paper, we study the property of the extension closure of some classes of objects in $(Tdownarrowmathscr{A})$, 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 $(Tdownarrowmathscr{A})$. Moreover, we characterize when special precovering classes in abelian categories $mathscr{A}$ and $mathscr{B}$ can induce special precovering classes in $(Tdownarrowmathscr{A})$. 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.
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