Complete Integral Closure and Noetherian Property for Integer-Valued Polynomial Rings

2017 ◽  
pp. 190-207
Keyword(s):  
Integral Closure ◽  
Polynomial Rings ◽  
Complete Integral Closure ◽  
Noetherian Property

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.

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