Ratliff–Rush Closures and Coefficient Modules

ISSN： 0021-8693

Source： Journal of Algebra, Vol.201, Iss.2, 1998-03, pp. : 584-603

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Abstract

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-module containing M. 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 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. Soc. 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 modules of M in F.