Essential prime divisors and projectively equivalent ideals

Katz, Daniel J.; McAdam, Stephen; Okon, James S.; Ratliff, Louis J.

doi:10.1016/0021-8693(87)90151-7

Cited by 12 publications

(4 citation statements)

References 6 publications

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“…(2.1.2) R denotes the integral closure of R in its total quotient ring. Concerning (2.1.6), Samuel introduced projectively equivalent ideals in 1952 in [19], and a number of properties of projective equivalence can be found in [4,5,[8][9][10][11]15,16]. Concerning (2.2.4), it is not true in general that integral closedness of ideals is preserved under a faithfully flat ring extension.…”

Section: Projectively Equivalent Idealsmentioning

confidence: 99%

Projectively full ideals in Noetherian rings

et al. 2006

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Let R be a Noetherian commutative ring with unit 1 = 0, and let I be a regular proper ideal of R. The set P(I ) of integrally closed ideals projectively equivalent to I is linearly ordered by inclusion and discrete. There is naturally associated to P(I ) a numerical semigroup S(I ); we have S(I ) = N if and only if every element of P(I ) is the integral closure of a power of the largest element J of P(I ). If this holds, the ideal J and the set P(I ) are said to be projectively full. If I is invertible and R is integrally closed, we prove that P(I ) is projectively full. We investigate the behavior of projectively full ideals in various types of ring extensions. We prove that a normal ideal I of a local ring (R, M) is projectively full if I M 2 and both the associated graded ring G(M) and the fiber cone ring F (I ) are reduced. We present examples of normal local domains (R, M) of altitude two for which the maximal ideal M is not projectively full.

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Section: Projectively Equivalent Idealsmentioning

confidence: 99%

Projectively full ideals in Noetherian rings

et al. 2006

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“…A number of properties of projective equivalence can be found in [7,8,[11][12][13][14]20,21]. In this section we explore the relation between projectively equivalent ideals and Rees valuation rings.…”

Section: Rees Valuation Rings and Projectively Equivalent Idealsmentioning

confidence: 99%

Projectively equivalent ideals and Rees valuations

et al. 2004

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Let R be a Noetherian ring. Two ideals I and J in R are projectively equivalent in case the integral closure of I i is equal to the integral closure of J j for some i, j ∈ N + . It is known that if I and J are projectively equivalent, then the set Rees I of Rees valuation rings of I is equal to the set Rees J of Rees valuation rings of J and the values of I and J with respect to these Rees valuation rings are proportional. We observe that the converse also holds. In particular, if the ideal I has only one Rees valuation ring V , then the ideals J projectively equivalent to I are precisely the ideals J such that Rees J = {V }. In certain cases such as: (i) dim R = 1, or (ii) R is a two-dimensional regular local domain, we observe that if I has more than one Rees valuation ring, then there exist ideals J such that Rees I = Rees J , but J is not projectively equivalent to I . If I and J are regular ideals of R, we prove that Rees I ∪ Rees J ⊆ Rees I J with equality holding if dim R 2, but not holding in general if dim R 3. We associate to I and to the set P(I ) of integrally closed ideals projectively equivalent to I a numerical semigroup S(I ) ⊆ N such that S(I ) = N if and only if there exists J ∈ P(I ) for which P(I ) = {(J n ) a | n ∈ N + }.

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“…Recall that an ideal J is projectively equivalent to I if (I m ) a = ( J n ) a for some positive integers m and n. In addition to the papers [2][3][4] and the references listed there, further interesting results concerning projective equivalence can be found in [6, Proposition 2.1], [7][8][9]. We also use the following definition, see [10] and [20, p. 111].…”

Section: Introductionmentioning

confidence: 97%

Bases of ideals and Rees valuation rings

Heinzer¹,

Ratliff²,

Rush³

2010

Journal of Algebra

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Let I be a regular proper ideal in a Noetherian ring R. We prove that there exists a simple free integral extension ring A of R such that the ideal I A has a Rees-good basis; that is, a basis c 1 , . . . , c g such that c i W = I W for i = 1, . . . , g and for all Rees valuation rings W of I A. Moreover, A may be constructed so that: (i) I A and I have the same Rees integers (with possibly different cardinalities), and (ii) A P is unramified over R P ∩R for each asymptotic prime divisor P of I A. Indeed, if H is a regular ideal in R such that each asymptotic prime divisor of H is contained in an asymptotic prime divisor of I, then (ii) holds for H A. If Card(Rees H) Card(Rees I), we prove that (i) also holds for H A and H. If I = (b 1 , . . . , b g )R and b 1 , . . . , b g is an asymptotic sequence, we prove that b 1 , . . . , b g is a Rees-good basis of I.

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Essential prime divisors and projectively equivalent ideals

Cited by 12 publications

References 6 publications

Projectively full ideals in Noetherian rings

Projectively full ideals in Noetherian rings

Projectively equivalent ideals and Rees valuations

Bases of ideals and Rees valuation rings

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