Contracting endomorphisms and Gorenstein modules

. A finite module M over a noetherian local ring R is said to be Gorenstein if Ext i ( k, M ) = 0 for all i ≠ dim R . An endomorphism φ: R → R of rings is called contracting if for some i ≥ 1. Letting φ R denote the R -module R with action induced by φ, we prove: A finite R -module M is Gorenstein if and only if Hom R ( φ R, M) ≅ M and Ext i R ( φ R, M) = 0 for 1 ≤ i ≤ depth R .