Let R be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated R-modules. For any n 0, we prove that R is a Gorenstein ring with self-injective dimension at most n if and only if every finitely generated left R-module and every finitely generated right R-module have torsionfree dimension at most n, if and only if every finitely generated le...

In this paper, we investigate the change of rings theorems for the Gorenstein dimensions over arbitrary rings. Namely, by the use of the notion of strongly Gorenstein modules, we extend the well-known first, second, and third change of rings theorems for the classical projective and injective dimensions to the Gorenstein projective and injective dimensions, respectively. Each of the results est...

We show the existence of a first order theory Cmd,e whose Noetherian models are precisely the local Cohen-Macaulay rings of dimension d and multiplicity e. The completion of a model of Cmd,e is again a model and is moreover Noetherian. If R is an equicharacteristic local Gorenstein ring of dimension d and multiplicity e with algebraically closed residue field and if the Artin Approximation Prop...

Let R be a local Noetherian domain of positive characteristic. A theorem of Hochster and Huneke (1992) states that if R is excellent, then the absolute integral closure of R is a big Cohen-Macaulay algebra. We prove that if R is the homomorphic image of a Gorenstein local ring, then all the local cohomology (below the dimension) of such a ring maps to zero in a finite extension of the ring. The...

We study the “q-commutative” power series ring R := kq[[x1, . . . , xn]], defined by the relations xixj = qijxjxi, for mulitiplicatively antisymmetric scalars qij in a field k. Our results provide a detailed account of prime ideal structure for a class of noncommutative, complete, local, noetherian domains having arbitrarily high (but finite) Krull, global, and classical Krull dimension. In par...