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Let $A$ be a right Noetherian positively graded ring. Let $Gr(A)$ be the category of right graded $A$-modules, and $Tors(A)$ be the full subcategory of $Gr(A)$ of torsion modules. Let $QGr(A)$ be the quotient category $Gr(A)/Tors(A)$ and $\pi:Gr(A)\to QGr(A)$ the canonical projection. It is known that $\pi$ preserves injective envelopes.

Does $\pi$ preserves essential extensions? Any comments will be grateful.

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