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Ratliff–Rush Closures and Coefficient Modules

✍ Scribed by Jung-Chen Liu

Book ID
102576158
Publisher
Elsevier Science
Year
1998
Tongue
English
Weight
239 KB
Volume
201
Category
Article
ISSN
0021-8693

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✦ Synopsis

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. w Ž . In analogy with a result of Ratliff and Rush Indiana Uni¨. Math. J. 27 1978 , x 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 FrM has finite length as an R-module. Then we define the Buchsbaum᎐Rim polynomial of M in F. In analogy w Ž .

x 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.

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✍ L.J. Ratliff Jr; D.E. Rush 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 812 KB

Formulas arc obtained for the determinants of certain matrices whose entries are zero and either binomial coefficients or their negatives. A consequence is that. for all integers II ?: 2 and k ;~2, there exists an (II -I)(k -I) x (n -I)(k -1) matrix M(n, k) whose entries arc the alternating binomial