Virtually Gorenstein Artin algebras are weakly Gorenstein

We prove that all semi-Gorenstein-projective modules over a virtually Gorenstein Artin algebra are Gorenstein projective; this answers affirmatively a question proposed by Ringel and Zhang in [25]. It turns out that the Auslander-Gorenstein Conjecture, the (Strong) Nakayama Conjecture and Tachikawa's First Conjecture hold for virtually Gorenstein Artin algebras. We also establish a new criterion for determining the projective (injective) dimensions of modules over virtually Gorenstein Artin algebras. This allows us to show the validity of the Auslander-Reiten Conjecture for (infinitely generated) modules over Artin algebras such that all the Gorenstein-projective modules are projective.