Gorenstein modules under Frobenius extensions
Bull. Korean Math. Soc. 2020 Vol. 57, No. 6, 1567-1579
https://doi.org/10.4134/BKMS.b200056
Published online September 4, 2020
Printed November 30, 2020
Fangdi Kong, Dejun Wu
Lanzhou University of Technology; Lanzhou University of Technology
Abstract : Let $R\subset S$ be a Frobenius extension of rings and $M$ a left $S$-module and let $\mathcal{X}$ be a class of left $R$-modules and $\mathcal{Y}$ a class of left $S$-modules. Under some conditions it is proven that $M$ is a $\mathcal{Y}$-Gorenstein left $S$-module if and only if $M$ is an $\mathcal{X}$-Gorenstein left $R$-module if and only if $\tp{S}{M}$ and $\Hom{S}{M}$ are $\mathcal{Y}$-Gorenstein left $S$-modules. This statement extends a known corresponding result. In addition, the situations of Ding modules, Gorenstein AC modules and projectively coresolved Gorenstein flat modules are considered under Frobenius extensions.
Keywords : Frobenius extension, $\mathcal{X}$-Gorenstein module, super finitely presented module, Ding module, PGF-module
MSC numbers : Primary 13B02, 16G50, 18G25
Supported by : This work was financially supported by NSF of China grants 11761047 and 11861043
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