Strongly Gorenstein flat and Gorenstein FP-injective modules. (Q2839024)

Strongly Gorenstein flat \(R\)-modules and Gorenstein FP-injective modules have been introduced and studied by \textit{N.-Q. Ding} et al. [J. Aust. Math. Soc. 86, No. 3, 323-338 (2009; Zbl 1200.16010)] and \textit{L. Mao} and \textit{N.-Q. Ding} [J. Algebra Appl. 7, No. 4, 491-506 (2008; Zbl 1165.16004)]. The study of homological properties of these concepts is continued in this paper. The concepts of the global strongly Gorenstein flat (resp. the global Gorenstein FP-injective) dimension of a ring \(R\) is introduced as the supremum of the strong Gorenstein flat (resp. Gorenstein FP-injective) dimensions of all \(R\)-modules. These two global dimensions are found to be equal if \(R\) is an \(m\)-FC ring or if \(R\) is a commutative coherent ring. In the case of a commutative Noetherian ring \(R\) of finite Krull dimension, it is proved that the strong Gorenstein flat dimension of an \(R\)-module \(M\) is finite if and only if the Gorenstein flat dimension of \(M\) is finite.