Let $A$ and $B$ be two finite-dimensional algebras over an algebraically closed field. Suppose that $A$ and $B$ are stably equivalent of Morita type; we prove that $A$ satisfies the Auslander–Reiten conjecture (resp. Gorenstein projective conjecture, strong Nakayama conjecture, Auslander–Gorenstein conjecture, Nakayama conjecture, Gorenstein symmetric conjecture) if and only if $B$ does so. This can provide new classes of algebras satisfying homological conjectures, and we give an example to illustrate it.

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.