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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 ${\mathrm{𝒢ℐ}}_C\left(R\right)\cap {𝒜}_C\left(R\right)$ is closed under direct sums and direct limits. Also, it is proved that under the above hypotheses the class ${\mathrm{𝒢ℐ}}_C\left(R\right)\cap {𝒜}_C\left(R\right)$ is covering....

Let $R$ be a commutative Noetherian ring. It is shown that the finitely generated $R$-module $M$ with finite Gorenstein dimension is reflexive if and only if $M_{𝔭}$ is reflexive for $𝔭\in \mathrm{Spec}\left(R\right)$ with $\mathrm{depth}\left(R_{𝔭}\right)\le 1$, and ${\text{G-}\text{dim}}_{R_{𝔭}}\left(M_{𝔭}\right)\le \mathrm{depth}\left(R_{𝔭}\right)-2$ for $𝔭\in \mathrm{Spec}\left(R\right)$ with $\mathrm{depth}\left(R_{𝔭}\right)\ge 2$. This gives a generalization of Serre and Samuel’s results on reflexive modules over a regular local ring and a generalization of a recent result due to Belshoff. In addition, for $n\ge 2$ we give a characterization of $n$-Gorenstein rings via Gorenstein dimension of the dual of modules. Finally it is shown...