Ratliff–Rush Closures and Coefficient Modules

Let (R, m) be ad-dimensional Noetherian local domain. SupposeMis a finitely generated torsion-freeR-module and supposeFis a freeR-module containingM. In analogy with a result of Ratliff and Rush [Indiana Univ. Math. J.27(1978), 929–934] concerning ideals, we define and prove existence and uniqueness of theRatliff–RushclosureofMinF. We also discuss properties of Ratliff–Rush closure. In addition to the preceding assumptions, supposeF/Mhas finite length as anR-module. Then we define theBuchsbaum–RimpolynomialofMinF. In analogy with the work of K. Shah [Trans. Amer. Math. Soc.327(1991), 373–384], we definecoefficientmodulesofMinF. Under the assumption thatRis quasi-unmixed, we prove existence and uniqueness of coefficient modules ofMinF.