Abstract In this paper, we generalize, with respect to the w-operation, the notions of FI-injective and FI-flat modules introduced by L. Mao and N. Ding. Let R be a ring. An R-module M is called w-FI-flat if Tor1R(E,M)=0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathrm{Tor}_{1}^{R}(E,M)=0$$end{document} for any absolutely w-pure R-module E, and M is said to be w-FI-injective if ExtR1(E,M)=0documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$$mathrm{Ext}_{R}^{1}(E,M)=0$$end{document} for any absolutely w-pure R-module E. The introduced modules are studied and compared with other classical modules. Among other things, we show that every module has a w-FI-flat cover. We use also the introduced modules to characterize some well known kind of rings.
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