RATLIFF-RUSH CLOSURES AND COEFFICIENT MODULES

Let (R, m) be a d-dimensional Noetherian local domain. Suppose M is a finitely generated torsion-free R-module and suppose F is a free R-mod ule containing M. In analogy with a result of Ratliff and Rush [Indian a Univ. Math. J. 27 (1978), 929-934] concerning ideals, we define and prove existence and uniqueness of the Ratliff-Rush closure of M in F. We also discuss properties of Ratliff-Rush closure. In addition to the preceding assumptions, suppose F/M has finite length as an R-module. Then we define the Buchsbaum-Rim polynomial of M in F. In analogy with the work of K. Shah [Trans. Amer. Math. Sec. 327 (1991), 373-384], we define coefficient modules of M in F. Under the assumption that R is quasi-unmixed, we prove existence and uniqueness of coefficient module s of M in F. (C) 1998 Academic Press.