Prime Ideals in GCD-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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On Nagata's Result about Height One Maximal Ideals and Depth One Minimal Prime Ideals (I)

Journal of the Indian Mathematical Society ◽

10.18311/jims/2018/20123 ◽

2018 ◽

Vol 85 (3-4) ◽

pp. 356

Author(s):

Paula Kemp ◽

Louis J. Ratliff, Jr. ◽

Kishor Shah

Keyword(s):

Integral Closure ◽

Prime Ideals ◽

Local Rings ◽

Finite Sets ◽

Maximal Ideals ◽

Valuation Rings ◽

Canonical Bijection ◽

Minimal Prime

It is shown that, for all local rings (R,M), there is a canonical bijection between the set DO(R) of depth one minimal prime ideals ω in the completion ^R of R and the set HO(R/Z) of height one maximal ideals ̅M' in the integral closure (R/Z)' of R/Z, where Z := Rad(R). Moreover, for the finite sets D := {V*/V* := (^R/ω)', ω ∈ DO(R)} and H := {V/V := (R/Z)'̅M', ̅M' ∈ HO(R/Z)}:(a) The elements in D and H are discrete Noetherian valuation rings.(b) D = {^V ∈ H}.

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Complete Integral Closure and Strongly Divisorial Prime Ideals

Communications in Algebra ◽

10.1081/agb-120023967 ◽

2003 ◽

Vol 31 (11) ◽

pp. 5447-5465 ◽

Cited By ~ 5

Author(s):

Valentina Barucci ◽

Stefania Gabelli ◽

Moshe Roitman

Keyword(s):

Integral Closure ◽

Prime Ideals ◽

Complete Integral Closure

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Minimal prime ideals of skew polynomial rings and near pseudo-valuation rings

Czechoslovak Mathematical Journal ◽

10.1007/s10587-013-0071-8 ◽

2013 ◽

Vol 63 (4) ◽

pp. 1049-1056

Author(s):

Vijay Kumar Bhat

Keyword(s):

Polynomial Rings ◽

Prime Ideals ◽

Skew Polynomial Rings ◽

Valuation Rings ◽

Minimal Prime ◽

Skew Polynomial

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Integral Closure of Noetherian Domains and Intersections of Rees Valuation Rings, (II)

Journal of the Indian Mathematical Society ◽

10.18311/jims/2017/6133 ◽

2017 ◽

Vol 84 (1-2) ◽

pp. 55

Author(s):

Paula Kemp ◽

Louis J. Ratliff, Jr. ◽

Kishor Shah

Keyword(s):

Valuation Ring ◽

Disjoint Union ◽

Integral Closure ◽

Primary Ideal ◽

Unique Factorization ◽

Unique Factorization Domain ◽

Valuation Rings ◽

Primary Ideals ◽

Rees Valuation

Let 1 < s1 < . . . < sk be integers, and assume that κ ≥ 2 (so sk ≤ 3). Then there exists a local UFD (Unique Factorization Domain) (R,M) such that:(1) Height(M) = sk.(2) R = R' = ∩{VI (V,N) € Vj}, where Vj (j = 1, . . . , κ) is the set of all of the Rees valuation rings (V,N) of the M-primary ideals such that trd((V I N) I (R I M)) = sj - 1.(3) With V1, . . . , Vκ as in (2), V1 ∪ . . . Vκis a disjoint union of all of the Rees valuation rings of allof the M-primary ideals, and each M-primary ideal has at least one Rees valuation ring in each Vj .

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Power Series Rings Over Prüfer v-multiplication Domains. II

Canadian Mathematical Bulletin ◽

10.4153/cmb-2016-046-5 ◽

2017 ◽

Vol 60 (1) ◽

pp. 63-76

Author(s):

Gyu Whan Chang

Keyword(s):

Power Series ◽

Prime Ideals ◽

Power Series Ring ◽

Series Ring ◽

Krull Domain ◽

Valuation Domains ◽

Power Series Rings ◽

Complete Integral Closure ◽

Minimal Prime ◽

Type Power

AbstractLet D be an integral domain, X1(D) be the set of height-one prime ideals of D, {Xβ} and {Xα} be two disjoint nonempty sets of indeterminates over D, D[{Xβ}] be the polynomial ring over D, and D[{Xβ}][[{Xα}]]1 be the first type power series ring over D[{Xβ}]. Assume that D is a Prüfer v-multiplication domain (PvMD) in which each proper integral t-ideal has only finitely many minimal prime ideals (e.g., t-SFT PvMDs, valuation domains, rings of Krull type). Among other things, we show that if X1(D) = Ø or DP is a DVR for all P ∊ X1(D), then D[{Xβ}][[{Xα}]]1D−{0} is a Krull domain. We also prove that if D is a t-SFT PvMD, then the complete integral closure of D is a Krull domain and ht(M[{Xβ}][[{Xα}]]1) = 1 for every height-one maximal t-ideal M of D.

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Definitions of integral elements and quotient rings over non-commutative rings with identity

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700009174 ◽

1972 ◽

Vol 13 (4) ◽

pp. 433-446 ◽

Cited By ~ 4

Author(s):

T. W. Atterton

Keyword(s):

Associative Ring ◽

Chain Condition ◽

Integral Closure ◽

Commutative Case ◽

P 75 ◽

Integrally Closed ◽

Complete Integral Closure ◽

Image Position ◽

Quotient Rings ◽

Ascending Chain Condition

Let B be an associative ring with identity, A a subring of B containing the identity of B. If B is commutative then it is customary to define an element b of B to be integral over A if it satisties an equation of the form for some a1, a2, …, an A. This definition does not generalize readily to the case when B is non-commutative. Van der Waerden ([11], p. 75) defines b ∈ B to be integral over A if all powers of b belong to a finite A-module. This definition is quite satisfactory when A satisfies the ascending chain condition for left ideals, but in the general case this type of integrity is not necessarily transitive, even when B is commutative. Krull [6] calls an element b ∈ B which satisfies the above condition almost integral over A (but he only considers the commutative case). The subset Ā of B consisting of all almost integral elements over A is called the complete integral closure of A in B. If Ā = A, A is said to be completely integrally closed in B. More recently (in [3]), Gilmer and Heinzer (see also Bourbaki, [1]) have discussed these properties in the commutative case and have shown that the complete integral closure of A in B need not be completely integrally closed in B. If B is not commutative, the set A of elements of B almost integral over A, may not even form a ring. In [5] p. 122, Jacobson uses a definition equivalent to Van der Waerden's for the non-commutative case but the definition applies only for a very restricted class of rings.

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A Lipman’s type construction, glueings and complete integral closure

Nagoya Mathematical Journal ◽

10.1017/s0027763000001276 ◽

1989 ◽

Vol 113 ◽

pp. 99-119 ◽

Cited By ~ 5

Author(s):

Valentina Barucci

Keyword(s):

Jacobson Radical ◽

Integral Closure ◽

Prime Ideals ◽

Finitely Generated ◽

Blowing Up ◽

Complete Integral Closure ◽

Type Construction

Given a semilocal 1-dimensional Cohen-Macauly ring A, J. Lipman in [10] gives an algorithm to obtain the integral closure Ā of A, in terms of prime ideals of A. More precisely, he shows that there exists a sequence of rings A = A0 ⊂ A1 ⊂… ⊂ Ai ⊂…, where, for each i, i ≥ 0, Ai+1 is the ring obtained from Ai by “blowing-up” the Jacobson radical ℛ i of Ai+ i.e. Ai+l = ∪n(ℛin:ℛin). It turns out that ∪ {Ai;i≥0} = Ā (cf. [10, proof of Theorem 4.6]) and, if Ā is a finitely generated A-module, the sequence {Ai; i ≥ 0} is stationary for some m and Am = Ā, so that

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