The Direct and Inverse Limits of w-Modules

In this article, the direct and inverse limits of w-modules of a commutative ring R are discussed. We prove that, if {M-i} is a family of w-modules of R, then <(lim)under right arrow>M-i and (lim) under left arrowM(i) are both w-modules, and that if {M-i} is a family of GV-torsion-free R-modules, then (<(lim)under right arrow>M-i)(w) = <(lim)under right arrow>(M-i)w. An example is given to show that ((lim) under left arrowM(i))(w) = (lim) under left arrow (M-i)(w) may not hold for a family of GV-torsion-free R-modules {M-i}.