In this paper, we show the existence of copure injective preenvelopes over noetherian rings and copure flat preenvelopes over commutative artinian rings. We use this to characterize n-Gorenstein rings. As a consequence, if the full subcategory of strongly copure injective (respectively flat) modules over a left and right noetherian ring R has cokernels (respectively kernels), then R is 2-Gorens...

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions rarely come with practical and robust criteria for finiteness, even over commutative noetherian local rings. Indeed, over such a ring ...

The classical global and weak dimensions of rings play an important role in the theory of rings and have a great impact on homological and commutative algebra. In this paper, we define and study the Gorenstein homological dimensions of commutative rings (Gorenstein projective, injective, and flat dimensions of rings) which introduce a new theory similar to the one of the classical homological d...

In this paper we shall apply modules of generalized fractions to extend the notion of the grade of an ideal, and to obtain characterizations of Gorenstein rings.

Unlike the Gorenstein projective and injective dimensions, the majority of results on the Gorenstein flat dimension have been established only over Noetherian (or coherent) rings. Naturally, one would like to generalize these results to any associative ring. In this direction, we show that the Gorenstein flat dimension is a refinement of the classical flat dimension over any ring; and we invest...

We introduce the notion of strongly ω -Gorenstein modules, where ω is a faithfully balanced self-orthogonal module. This gives a common generalization of both Gorenstein projective (injective) modules and ω-Gorenstein modules. We investigate some characterizations of strongly ω -Gorenstein modules. Consequently, some properties under change of rings are obtained. Keywords—faithfully balanced se...