Gorenstein invariants under right Quasi-Frobenius extensions

The focus of this work was to establish relations between Gorenstein projective modules linked by right quasi-Frobenius extensions of rings. As applications, the right quasi-Frobenius extensions of (weakly) Gorenstein algebras, Cohen-Macaulay finite algebras and the Cohen-Macaulay free algebra were studied. Suppose that $\mathrm{\Gamma }$ was a right quasi-Frobenius extension of an Artin algebra $\mathrm{\Lambda }$ with $\mathrm{\Gamma }$ a completely faithful left $\mathrm{\Lambda }$-module. We demonstrated that if $\mathrm{\Gamma }$ exhibited (weakly) Gorenstein properties, then $\mathrm{\Lambda }$ did so. Additionally, under the condition of $\mathrm{\Gamma }$ being a separable extension of $\mathrm{\Lambda }$, the converse became valid.