ABSTRACT
By definition, the Gorenstein projective (injective) modules are the cycles of the totally acyclic complexes of projective (injective) modules. The Gorenstein flat modules are the cycles of the F-totally acyclic complexes. It is a natural question to consider whether or not these conditions actually characterize Gorenstein rings, or more generally, whether or not it is possible to characterize Gorenstein rings in terms of acyclic complexes of (Gorenstein) injectives, (Gorenstein) projectives, and (Gorenstein) flats. This is the main question considered in this chapter. It considers the question over noncommutative rings and a two-sided noetherian ring R such that every acyclic complex of injective modules is totally acyclic. The chapter proves that if, furthermore, R satisfies the Auslander condition and has finite finitistic flat dimension, then every injective R-module has finite flat dimension. This result is used to prove the characterization of Iwanaga-Gorenstein rings.
