Power series over generalized Krull domains

Paran, Elad; Temkin, Michael

doi:10.1016/j.jalgebra.2009.08.011

Cited by 13 publications

(7 citation statements)

References 5 publications

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“…and (iii) F has finite character; that is, if x ∈ K is nonzero, then x is a nonunit in only finitely many valuation rings of F. This class of rings has been studied by a number of authors; see for example [13,14,19,30,31,32]. In our setting, when S S * the ring S * is a generalized Krull domain whose defining family F consists of rank 1 valuation rings such that all but at most one member (namely, W ) is a DVR.…”

Section: The Complete Integral Closure Of a Shannon Extensionmentioning

confidence: 99%

Ideal theory of infinite directed unions of local quadratic transforms

Heinzer¹,

Loper²,

Olberding³

et al. 2017

Journal of Algebra

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Let (R, m) be a regular local ring of dimension at least 2. Associated to each valuation domain birationally dominating R, there exists a unique sequence {R n } of local quadratic transforms of R along this valuation domain. We consider the situation where the sequence {R n } n≥0 is infinite, and examine ideal-theoretic properties of the integrally closed local domain S = n≥0 R n . Among the set of valuation overrings of R, there exists a unique limit point V for the sequence of order valuation rings of the R n . We prove the existence of a unique minimal proper Noetherian overring T of S, and establish the decomposition S = T ∩ V . If S is archimedian, then the complete integral closure S * of S has the form S * = W ∩ T , where W is the rank 1 valuation overring of V .

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Section: The Complete Integral Closure Of a Shannon Extensionmentioning

confidence: 99%

Ideal theory of infinite directed unions of local quadratic transforms

Heinzer¹,

Loper²,

Olberding³

et al. 2017

Journal of Algebra

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show abstract

“…Gilmer in [3, page 524] defines an integral domain A with quotient field K to be a generalized Krull domain if there is a set F of rank 1 valuation overrings of A such that: (i) A = V∈F V; (ii) for each (V, M V ) ∈ F, we have V = A M V ∩A ; and (iii) F has finite character; that is, if x ∈ K is nonzero, then x is a nonunit in only finitely many valuation rings of F. The class of generalized Krull domains has been studied by a number of authors; see for example [7,8,12,18,19,20]. Proof.…”

Section: The Complete Integral Closure Of An Archimedean Shannon Extementioning

confidence: 99%

Asymptotic properties of infinite directed unions of local quadratic transforms

2017

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Let (R, m) be a regular local ring of dimension at least 2. For each valuation domain birationally dominating R, there is an associated sequence {R n } of local quadratic transforms of R. We consider the case where this sequence {R n } n≥0 is infinite and examine properties of the integrally closed local domain S = n≥0 R n in the case where S is not a valuation domain. For this sequence, there is an associated boundary valuation ring V = n≥0 i≥n V i , where V i is the order valuation ring of R i . There exists a unique minimal proper Noetherian overring T of S. T is the regular Noetherian UFD obtained by localizing outside the maximal ideal of S and S = V ∩ T . In the present paper, we define functions w and e, where w is the asymptotic limit of the order valuations and e is the limit of the orders of transforms of principal ideals. We describe V explicitly in terms of w and e and prove that V is either rank 1 or rank 2. We define an invariant τ associated to S that is either a positive real number or +∞. If τ is finite, then S is archimedean and T is not local. In this case, the function w defines the rank 1 valuation overring W of V and W dominates S. The rational dependence of τ over w(T × ) determines whether S is completely integrally closed and whether V has rank 1. We give examples where S is completely integrally closed. If τ is infinite, then S is non-archimedean and T is local. In this case, the function e defines the rank 1 valuation overring E of V . The valuation ring E is a DVR and E dominates T , and in certain cases we prove that E is the order valuation ring of T .

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“…A domain R is a rational generalized Krull domain if and only if R • is a rational generalized Krull monoid (for recent work on these kinds of domains, see [42,47]). A v-Marot ring (and, in particular, a domain) R is a Krull ring if and only if its multiplicative monoid of regular elements R • is a Krull monoid ([33, Theorem 3.5]), and we set C(R) = C(R • ).…”

Section: Proposition 21 (Arithmetic Of Tame Monoids)mentioning

confidence: 99%

A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems

et al. 2014

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Abstract. Let R be a ring and let C be a small class of right R-modules which is closed under finite direct sums, direct summands, and isomorphisms. Let V(C) denote a set of representatives of isomorphism classes in C and, for any module M in C, let [M ] denote the unique element in V(C) isomorphic to M . Then V(C) is a reduced commutative semigroup with operation defined by [M ] , and this semigroup carries all information about direct-sum decompositions of modules in C. This semigroup-theoretical point of view has been prevalent in the theory of direct-sum decompositions since it was shown that if End R (M ) is semilocal for all M ∈ C, then V(C) is a Krull monoid. Suppose that the monoid V(C) is Krull with a finitely generated class group (for example, when C is the class of finitely generated torsion-free modules and R is a one-dimensional reduced Noetherian local ring). In this case we study the arithmetic of V(C) using new methods from zero-sum theory. Furthermore, based on module-theoretic work of Lam, Levy, Robson, and others we study the algebraic and arithmetic structure of the monoid V(C) for certain classes of modules over Prüfer rings and hereditary Noetherian prime rings.

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Power series over generalized Krull domains

Cited by 13 publications

References 5 publications

Ideal theory of infinite directed unions of local quadratic transforms

Ideal theory of infinite directed unions of local quadratic transforms

Asymptotic properties of infinite directed unions of local quadratic transforms

A semigroup-theoretical view of direct-sum decompositions and associated combinatorial problems

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