Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a commutative ring. An $R$-module $M$ is said to be $w$-flat if $\Tor^{R}_{1}(M,N)$ is $GV$-torsion for any $R$-module $N$. It is known that every flat module is $w$-flat, but the converse is not true in general. The $w$-flat dimension of a module is defined in terms of $w$-flat resolutions. In this paper, we study the $w$-flat dimension of an injective $w$-module. To do so, we introduce and study the so-called $w$-copure (resp., strongly $w$-copure) flat modules and the $w$-copure flat dimensions for modules and rings. The relations between the introduced dimensions and other (classical) homological dimensions are discussed. We also study change of rings theorems for the $w$-copure flat dimension in various contexts. Finally some illustrative examples regarding the introduced concepts are given.

Keywords: $w$-copure flat module, strongly $w$-copure flat module, $w$-copure flat dimension, $w$-linked

MSC numbers: 13D05, 13D07, 13H05

Supported by: This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1D1A3B03033342).