Commutative Noetherian rings possessing a homological property

For a commutative Noetherian local ring A, we have the following proposition: A is aGorenstein ring if and only if for all finitely generated A-modules M, id_AM~-is finite if andonly if pd_AM is finite. Now, we consider the following property: given a Noetherian localring A, for an arbitrary finitely generated A-module M, id_AM is finite, implying that pd_AMis finite. We ask: what ring is characterized by the above property? In this note, we firstconsider the above question; then for commutative Noetherian ring A (not necessarily lo-