Let [Formula: see text] be an integer. We introduce the notions of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Then we investigate the properties of these modules by using the properties of special finitely presented graded modules and obtain some equivalent characterizations of [Formula: see text]-gr-coherent rings in terms of [Formula: see text]-FP-gr-injective and [Formula: see text]-gr-flat modules. Moreover, we prove that the pairs (gr-[Formula: see text], gr-[Formula: see text]) and (gr-[Formula: see text], gr-[Formula: see text]) are duality pairs over left [Formula: see text]-coherent rings, where gr-[Formula: see text] and gr-[Formula: see text] denote the subcategories of [Formula: see text]-FP-gr-injective left [Formula: see text]-modules and [Formula: see text]-gr-flat right [Formula: see text]-modules respectively. As applications, we obtain that any graded left (respectively, right) [Formula: see text]-module admits an [Formula: see text]-FP-gr-injective (respectively, [Formula: see text]-gr-flat) cover and preenvelope.
A ring R is called Gorenstein hereditary (G-hereditary) if every submodule of a projective module is Gorenstein projective (i.e. Ggldim (R) ≤ 1). In this paper, we settle a question raised by Mahdou and Tamekkante in [On (strongly) Gorenstein (semi)hereditary rings, Arab. J. Sci. Eng.36 (2011) 436] about the coherence of G-hereditary rings. It is shown that a ring R is Gorenstein semihereditary if and only if every finitely generated submodule of a projective module is Gorenstein projective. As a consequence of this result, we have that every G-hereditary ring R is coherent.
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