Ratliff-Rush Closures and Coefficient Modules

Jung Chen Liu

doi:10.1006/jabr.1997.7300

Ratliff-Rush Closures and Coefficient Modules

Jung Chen Liu^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

Abstract

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.

Original language	English
Pages (from-to)	584-603
Number of pages	20
Journal	Journal of Algebra
Volume	201
Issue number	2
DOIs	https://doi.org/10.1006/jabr.1997.7300
Publication status	Published - 1998 Mar 15
Externally published	Yes

Keywords

Buchsbaum-Rim multiplicity
Coefficient ideal
Hilbert polynomial
Integral closure
Ratliff-Rush closure
Reduction of a module
Reduction of an ideal
Torsion-free symmetric algebra

ASJC Scopus subject areas

Algebra and Number Theory

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10.1006/jabr.1997.7300

Cite this

@article{24594311832540d98b37cf2013b3aa78,

title = "Ratliff-Rush Closures and Coefficient Modules",

abstract = "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.",

keywords = "Buchsbaum-Rim multiplicity, Coefficient ideal, Hilbert polynomial, Integral closure, Ratliff-Rush closure, Reduction of a module, Reduction of an ideal, Torsion-free symmetric algebra",

author = "Liu, {Jung Chen}",

note = "Funding Information: * E-mail address: [email protected]. ²The author was partially supported by the Purdue Research Foundation.",

year = "1998",

month = mar,

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doi = "10.1006/jabr.1997.7300",

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journal = "Journal of Algebra",

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AU - Liu, Jung Chen

N1 - Funding Information: * E-mail address: [email protected]. ²The author was partially supported by the Purdue Research Foundation.

PY - 1998/3/15

Y1 - 1998/3/15

N2 - 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.

AB - 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.

KW - Buchsbaum-Rim multiplicity

KW - Coefficient ideal

KW - Hilbert polynomial

KW - Integral closure

KW - Ratliff-Rush closure

KW - Reduction of a module

KW - Reduction of an ideal

KW - Torsion-free symmetric algebra

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DO - 10.1006/jabr.1997.7300

M3 - Article

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SN - 0021-8693

VL - 201

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EP - 603

JO - Journal of Algebra

JF - Journal of Algebra

IS - 2

ER -

Ratliff-Rush Closures and Coefficient Modules

Abstract

Keywords

ASJC Scopus subject areas

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