Homological Properties of (Graded) Noetherian PI Rings

Let R be a connected, graded, Noetherian PI ring. If injdim( R) = n < ∞, then we prove that R is Auslander-Gorenstein and Cohen-Macaulay, with Gelfand-Kirillov dimension equal to n. If gldim( R) = n < ∞, then R is a domain, finitely generated as a module over its centre and a maximal order in its quotient division ring. Similar results hold if R is assumed to be local rather than connected graded. Alternatively, suppose that R is a Noetherian PI ring with gldim( R) < ∞ such that hd( R/ M 1) = hd( R/ M 2) for any two maximal ideals M i in the same clique. Then, R is a direct sum of prime rings, is integral over its centre, and is Auslander-Gorenstein. If R is a prime ring, then the centre Z( R) of R is a Krull domain and R equals its trace ring TR. Moreover, hd( R/ M) = height( M), for every maximal ideal M of R.