$n$-Gorenstein Projective Modules and Dimensions over Frobenius Extensions
Received:February 16, 2020  Revised:September 26, 2020
Key Word: Frobenius extensions   $n$-Gorenstein projective modules   $n$-Gorenstein projective dimensions
Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant No.11561061).
 Author Name Affiliation Miao WANG Department of Mathematics, Northwest Normal University, Gansu 730070, P. R. China Zhanping WANG Department of Mathematics, Northwest Normal University, Gansu 730070, P. R. China
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In this paper, we study $n$-Gorenstein projective modules over Frobenius extensions and $n$-Gorenstein projective dimensions over separable Frobenius extensions. Let $R\subset A$ be a Frobenius extension of rings and $M$ any left $A$-module. It is proved that $M$ is an $n$-Gorenstein projective left $A$-module if and only if $A\otimes_{R}M$ and $\mathrm{Hom}_{R}(A,M)$ are $n$-Gorenstein projective left $A$-modules if and only if $M$ is an $n$-Gorenstein projective left $R$-module. Furthermore, when $R\subset A$ is a separable Frobenius extension, $n$-Gorenstein projective dimensions are considered.