Geodesic
Relative Gorenstein injective covers with respect to a semidualizing module
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 87-95
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Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_{C}$-injective module $G$, the character module $G^{+}$ is $G_{C}$-flat, then the class $\mathcal {GI}_{C}(R)\cap \mathcal {A}_{C}(R)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class $\mathcal {GI}_{C}(R)\cap \mathcal {A}_{C}(R)$ is covering.
Let $R$ be a commutative Noetherian ring and let $C$ be a semidualizing $R$-module. We prove a result about the covering properties of the class of relative Gorenstein injective modules with respect to $C$ which is a generalization of Theorem 1 by Enochs and Iacob (2015). Specifically, we prove that if for every $G_{C}$-injective module $G$, the character module $G^{+}$ is $G_{C}$-flat, then the class $\mathcal {GI}_{C}(R)\cap \mathcal {A}_{C}(R)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class $\mathcal {GI}_{C}(R)\cap \mathcal {A}_{C}(R)$ is covering.
DOI : 10.21136/CMJ.2017.0379-15
Classification : 13D05, 13D45, 18G20
Keywords: semidualizing module; $G_{C}$-flat module; $G _{C}$-injective module; cover; envelope 
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Tavasoli, Elham; Salimi, Maryam. Relative Gorenstein injective covers with respect to a semidualizing module. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 87-95. doi: 10.21136/CMJ.2017.0379-15

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