On the Complete Integral Closure of the Rees Algebra

2019 ◽  
pp. 207-222
Author(s):  
Stefania Gabelli ◽  
Anna Guerrieri
Keyword(s):  
Integral Closure ◽  
Rees Algebra ◽  
Complete Integral Closure

scholarly journals On complete integral closure and Archimedean valuation domains

1996 ◽  
Vol 61 (3) ◽  
pp. 377-380 ◽  
Author(s):  
Robert Gilmer
Keyword(s):  
Integral Domain ◽  
Integral Closure ◽  
Extension Field ◽  
Quotient Field ◽  
Valuation Domains ◽  
Complete Integral Closure

AbstractSuppose D is an integral domain with quotient field K and that L is an extension field of K. We show in Theorem 4 that if the complete integral closure of D is an intersection of Archimedean valuation domains on K, then the complete integral closure of D in L is an intersection of Archimedean valuation domains on L; this answers a question raised by Gilmer and Heinzer in 1965.

scholarly journals Complete Integral Closure and Strongly Divisorial Prime Ideals

2003 ◽  
Vol 31 (11) ◽  
pp. 5447-5465 ◽  
Author(s):  
Valentina Barucci ◽  
Stefania Gabelli ◽  
Moshe Roitman
Keyword(s):  
Integral Closure ◽  
Prime Ideals ◽  
Complete Integral Closure

scholarly journals ON THE ARITHMETIC OF MORI MONOIDS AND DOMAINS

2019 ◽  
Vol 62 (2) ◽  
pp. 313-322 ◽  
Author(s):  
QINGHAI ZHONG
Keyword(s):  
Integral Closure ◽  
Class Groups ◽  
Sets Of Lengths ◽  
Complete Integral Closure

AbstractLet R be a Mori domain with complete integral closure $\widehat R$, nonzero conductor $\mathfrak f = (R: \widehat R)$, and suppose that both v-class groups ${{\cal C}_v}(R)$ and ${{\cal C}_v}(3\widehat R)$ are finite. If $R \mathfrak f$ is finite, then the elasticity of R is either rational or infinite. If $R \mathfrak f$ is artinian, then unions of sets of lengths of R are almost arithmetical progressions with the same difference and global bound. We derive our results in the setting of v-noetherian monoids.

Prime Ideals in GCD-Domains

1974 ◽  
Vol 26 (1) ◽  
pp. 98-107 ◽  
Author(s):  
Philip B. Sheldon
Keyword(s):  
General Setting ◽  
Chain Condition ◽  
Integral Closure ◽  
Prime Ideals ◽  
Unique Factorization Domain ◽  
Principal Ideals ◽  
Valuation Rings ◽  
Complete Integral Closure ◽  
Minimal Prime ◽  
Bezout Domains

A GCD-domain is a commutative integral domain in which each pair of elements has a greatest common divisor (g.c.d.). (This is the terminology of Kaplansky [9]. Bourbaki uses the term ''anneau pseudobezoutien" [3, p. 86], while Cohn refers to such rings as "HCF-rings" [4].) The concept of a GCD-domain provides a useful generalization of that of a unique factorization domain (UFD), since several of the standard results for a UFD can be proved in this more general setting (for example, integral closure, some properties of D[X], etc.). Since the class of GCD-domains contains all of the Bezout domains, and in particular, the valuation rings, it is clear that some of the properties of a UFD do not hold in general in a GCD-domain. Among these are complete integral closure, ascending chain condition on principal ideals, and some of the important properties of minimal prime ideals.

scholarly journals On Ratliff-Rush closure of modules

2020 ◽  
Vol 126 (2) ◽  
pp. 170-188
Author(s):  
Naoki Endo
Keyword(s):  
Integral Closure ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Projective Scheme

In this paper, we introduce the notion of Ratliff-Rush closure of modules and explore whether the condition of the Ratliff-Rush closure coincides with the integral closure. The main result characterizes the condition in terms of the normality of the projective scheme of the Rees algebra. In conclusion, we shall give a criterion for the Buchsbaum Rees algebras.

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