On Gorenstein flat dimension

Abstract

The theory of Gorenstein flat dimension is not complete since it is not yet known whether the category 𝒢ℱ(R) of Gorenstein flat modules over a ring R is projectively resolving or not. Besides, it arises from recent investigations on this subject that there exists several ways of measuring the Gorenstein flat dimension of modules which turn out to coincide with the usual one in the case where 𝒢ℱ(R) is projectively resolving. These alternate procedures yield new invariants which enjoy very nice behavior for an arbitrary ring R. In this paper, we introduce and study one of these invariants called the cover Gorenstein flat dimension of a module M and denoted by CGfd R(M). This new entity stems from a sort of a Gorenstein flat precover of M. First, for each R-module M, we prove that Gfd R(M) ≤ CGfd R(M) for each R-module M with [Formula: see text] whenever CGfd R(M) is finite. Also, we show that 𝒢ℱ(R) is projectively resolving if and only if the Gorenstein flat dimension and the introduced cover Gorenstein flat dimension coincide. In particular, if R is a right coherent ring, then CGfd R(M) = Gfd R(M) for any R-module M. As a consequence, we prove that if R is a left and right GF-closed, then the Gorenstein weak global dimension of R is left–right symmetric and it is related to the cohomological invariants leftsfli(R) and rightsfli(R) by the formula [Formula: see text]