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Abstract. In 1966, Auslander introduced the notion of the G-dimension of a finitely generated module over a Cohen-Macaulay noetherian ring and found the basic properties of these dimensions. His results were valid over a local Cohen-Macaulay ring admitting a dualizing module (also see Auslander and Bridger (Mem. Amer. Math. Soc., vol. 94, 1969)). Enochs and Jenda attempted to dualize the notion of G-dimensions. It seemed appropriate to call the modules with G-dimension 0 Gorenstein projective, so the basic problem was to define Gorenstein injective modules. These were defined in Math. Z. 220 (1995), 611-633 and were shown to have properties predicted by Auslander's results. The way we define Gorenstein injective modules can be dualized, and so we can define Gorenstein projective modules (i.e. modules of G-dimension 0) whether the modules are finitely generated or not. The investigation of these modules and also Gorenstein flat modules was continued by Enochs, Jenda, Xu and Torrecillas. However, to get good results it was necessary to take the base ring Gorenstein. H.-B. Foxby introduced a duality between two full subcategories in the category of modules over a local CohenMacaulay ring admitting a dualizing module. He proved that the finitely generated modules in one category are precisely those of finite G-dimension. We extend this result to modules which are not necessarily finitely generated and also prove the dual result, i.e. we characterize the modules in the other class defined by Foxby. The basic result of this paper is that the two classes involved in Foxby's duality coincide with the classes of those modules having finite Gorenstein projective and those having finite Gorenstein injective dimensions. We note that this duality then allows us to extend many of our results to the original Auslander setting.

A general framework for the design of feedback registers based on algebra over complete rings is described. These registers generalize linear feedback shift registers and feedback with carry shift registers. Basic properties of the output sequences are studied: relations to the algebra of the underlying ring; synthesis of the register from the sequence (which has implications for cryptanalysis); and basic statistical properties. These considerations lead to security measures for stream ciphers, analogous to the notion of linear complexity that arises from linear feedback shift registers. We also show that when the underlying ring is a polynomial ring over a finite field, the new registers can be simulated by linear feedback shift registers with small nonlinear filters.

Abstract. Let R be a commutative Noetherian ring, and let M be an Rmodule. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers µ i (p, M ) were defined for all primes p and all integers i ≥ 0 by use of the minimal injective resolution of M . It is well known thatOn the other hand, if M is finitely generated, the Betti numbers β i (p, M ) are defined by the minimal free resolution of Mp over the local ring Rp. In an earlier paper of the second author (1995), using the flat covers of modules, the invariants π i (p, M ) were defined by the minimal flat resolution of M over Gorenstein rings. The invariants π i (p, M ) were shown to be somehow dual to the Bass numbers. In this paper, we use homologies to compute these invariants and show thatfor any cotorsion module M . Comparing this with the computation of the Bass numbers, we see that Ext is replaced by Tor and the localization Mp is replaced by Hom R (Rp, M) (which was called the colocalization of M at the prime ideal p by Melkersson and Schenzel).

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