Homological Properties of (Graded) Noetherian PI Rings

Abstract

Let R be a connected, graded, Noetherian PI ring. If injdim(R) = n R is Auslander-Gorenstein and Cohen-Macaulay, with Gelfand-Kirillov dimension equal to n. If gldim(R) = n 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) R/M1) = hd(R/M2) for any two maximal ideals Mi 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.