Auslander-type conditions and weakly Gorenstein algebras

Let R be an Artin algebra. Under certain Auslander-type conditions, we give some equivalent characterizations of (weakly) Gorenstein algebras in terms of the properties of Gorenstein projective modules and modules satisfying Auslander-type conditions. As applications, we provide some support for several homological conjectures. In particular, we prove that if R is left quasi Auslander, then R is Gorenstein if and only if it is (left and) right weakly Gorenstein; and that if R satisfies the Auslander condition, then R is Gorenstein if and only if it is left or right weakly Gorenstein. This is a reduction of an Auslander--Reiten's conjecture, which states that R is Gorenstein if R satisfies the Auslander condition.