Abstract: ${\ell}$_n-flat module is defined and studied in this paper. A right R-module M is said to be ${\ell}$_n-flat provided that Tor^R₁(M, C)=0 for every n-cotorsion left R-module C. It is proved that M is flat if and only if M is ${\ell}$_n-flat and fd_RM≤1; and ${\ell}$_nF is closed with pure submodules and pure quotient modules. Moreover, C₁F and CF are identical to the class of flat modules, in addition, if R is domain and every C₂-flat module is also flat module. Finally, R has week global dimension ≤n if and only if the n th-syzygy of any right R-module is ${\ell}$_n-flat, and R is von Neumann regular ring if and only if every right module is ${\ell}$_n-flat module.

Key words: n-cotorsion module, ${\ell}$_n-injective module, ${\ell}$_n-flat module, weak global dimension, von Neumann regular rings