Connected graded Gorenstein algebras with enough normal elements

We generalize [12, 1.1 and 1.2] to the following situation. Theorem 1. Let A be a connected graded noetherian algebra of injective dimension d such that every nonsimple graded prime factor ring of A contains a homogeneous normal element of positive degree. Then: (1) A is Auslander-Gorenstein and Cohen-Macaulay. (2) A has a quasi-Frobenius quotient ring. (3) Every minimal prime ideal P is graded and GKdim A/P = d. (4) If, moreover, A has finite global dimension, then A is a domain and a maximal order in irs quotient division ring. To prove the above we need the following result, which is a generalization of [3, 2.46(ii)]. Theorem 2. Let A be a connected graded noetherian AS-Gorenstein algebra of injective dimension d. Then: (1) The last term of the minimal injective resolution of A(A) is isomorphic to a shift of A*. (2) For every noetherian graded A-module M, <(Ext)under bar (d)>(M, A) is finite dimensional over k. (C) 1997 Academic Press.