scholarly journals Symbolic powers of prime ideals with applications to hypersurface rings

1989 ◽  
Vol 113 ◽  
pp. 161-172 ◽  
Author(s):  
Sam Huckaba
Keyword(s):  
Prime Ideal ◽  
Complete Intersection ◽  
Fundamental Problem ◽  
Three Dimensional ◽  
Regular Sequence ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Commutative Noetherian Ring ◽  
Symbolic Powers ◽  
Associated Prime

Let R be a commutative Noetherian ring and suppose q is a prime ideal of R. A fundamental problem is to decide when powers qn of q are primary (that is qn is its own primary decomposition). If q is generated by a regular sequence then powers of q are always primary, because G(q, R) (the associated graded ring of R with respect to q) is an integral domain (see [12 page 98] and also [5 (2.1)]). Let qn) denote the nth symbolic power of q-defined by q(n) = {rεR|there exists sεR\q such that sr ε qn}. Then qn is primary if and only if qn = q(n) If q is generated by a regular sequence then we call it a complete intersection prime ideal, so if q is a complete intersection prime ideal then qn ≠ q(n) for all n ≥ 1. If q is not a complete intersection then powers need not be primary. If R is a three-dimensional regular local ring and q is a non-complete intersection height two prime ideal for example, then Huneke showed [11 Corollary (2.5)] that qn = q(n) for all n ≥ 2. Thus, for such a prime q it is impossible for qn to occur in the primary decomposition of any ideal. This phenomena increases the difficulty in finding a primary decomposition for an ideal having q as an associated prime.

scholarly journals Ratliff–Rush closures and linear growth of primary decompositions of ideals

2017 ◽  
Vol 16 (04) ◽  
pp. 1750068
Author(s):  
Monireh Sedghi
Keyword(s):  
Positive Integer ◽  
Noetherian Ring ◽  
Linear Growth ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Commutative Noetherian Ring ◽  
Associated Prime Ideals ◽  
Decomposition Formula ◽  
Associated Prime ◽  
Relevant Component

Let [Formula: see text] be a commutative Noetherian ring, [Formula: see text] a nonzero finitely generated [Formula: see text]-module and [Formula: see text] an ideal of [Formula: see text]. First purpose of this paper is to show that the sequences [Formula: see text] and [Formula: see text], [Formula: see text] of associated prime ideals are increasing and eventually stabilize. This extends the main result of Mirbagheri and Ratliff [On the relevant transform and the relevant component of an ideal, J. Algebra 111 (1987) 507–519, Theorem 3.1]. In addition, a characterization concerning the set [Formula: see text] is included. A second purpose of this paper is to prove that [Formula: see text] has linear growth primary decompositions for Ratliff–Rush closures with respect to [Formula: see text], that is, there exists a positive integer [Formula: see text] such that for every positive integer [Formula: see text], there exists a minimal primary decomposition [Formula: see text] in [Formula: see text] with [Formula: see text], for all [Formula: see text].

On asymptotic stability for sets of prime ideals connected with the powers of an ideal

1990 ◽  
Vol 107 (2) ◽  
pp. 267-271 ◽  
Author(s):  
Leif Melkersson
Keyword(s):  
Asymptotic Stability ◽  
Prime Ideal ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Attached Prime ◽  
Associated Prime ◽  
Secondary Representation

The notion of associated prime ideal and the related one of primary decomposition are classical. In a dual way one defines attached prime ideals and secondary representation. This theory is developed in the appendix to §6 in Matsumura[5] and in Macdonald[3].

Positive deissler rank and the complexity of injective modules

1988 ◽  
Vol 53 (1) ◽  
pp. 284-293 ◽  
Author(s):  
T. G. Kucera
Keyword(s):  
Prime Ideal ◽  
Lower Bounds ◽  
Upper Bound ◽  
Noetherian Ring ◽  
Krull Dimension ◽  
Primary Ideal ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Primary Ideals

This is the second of two papers based on Chapter V of the author's Ph.D. thesis [K1]. For acknowledgements please refer to [K3]. In this paper I apply some of the ideas and techniques introduced in [K3] to the study of a very specific example. I obtain an upper bound for the positive Deissler rank of an injective module over a commutative Noetherian ring in terms of Krull dimension. The problem of finding lower bounds is vastly more difficult and is not addressed here, although I make a few comments and a conjecture at the end.For notation, terminology and definitions, I refer the reader to [K3]. I also use material on ideals and injective modules from [N] and [Ma]. Deissler's rank was introduced in [D].For the most part, in this paper all modules are unitary left modules over a commutative Noetherian ring Λ but in §1 I begin with a few useful results on totally transcendental modules and the relation between Deissler's rank rk and rk+.Recall that if P is a prime ideal of Λ, then an ideal I of Λ is P-primary if I ⊂ P, λ ∈ P implies that λn ∈ I for some n and if λµ ∈ I, λ ∉ P, then µ ∈ I. The intersection of finitely many P-primary ideals is again P-primary, and any P-primary ideal can be written as the intersection of finitely many irreducible P-primary ideals since Λ is Noetherian. Every irreducible ideal is P-primary for some prime ideal P. In addition, it will be important to recall that if P and Q are prime ideals, I is P-primary, J is Q-primary, and J ⊃ I, then Q ⊃ P. (All of these results can be found in [N].)

Links of prime ideals

1994 ◽  
Vol 115 (3) ◽  
pp. 431-436 ◽  
Author(s):  
Alberto Corso ◽  
Claudia Polini ◽  
Wolmer V. Vasconcelos
Keyword(s):  
Commutative Algebra ◽  
Noetherian Ring ◽  
Duality Theory ◽  
Regular Sequence ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Polynomial Ideals ◽  
Common Operation

Roughly speaking, a link of an ideal of a Noetherian ring R is an ideal of the form I = (z): , where z = z1, …, zg is a regular sequence and g is the codimension of . This is a very common operation in commutative algebra, particularly in duality theory, and plays an important role in current methods to effect primary decomposition of polynomial ideals (see [2]).

scholarly journals Values and bounds for the Stanley depth

2011 ◽  
Vol 27 (2) ◽  
pp. 217-224
Author(s):  
MUHAMMAD ISHAQ ◽  
Keyword(s):  
Prime Ideal ◽  
Polynomial Algebra ◽  
Monomial Ideal ◽  
Prime Ideals ◽  
Stanley Depth ◽  
Stanley Conjecture ◽  
Associated Prime Ideals ◽  
Disjoint Sets ◽  
Associated Prime

We give different bounds for the Stanley depth of a monomial ideal I of a polynomial algebra S over a field K. For example we show that the Stanley depth of I is less than or equal to the Stanley depth of any prime ideal associated to S/I. Also we show that the Stanley conjecture holds for I and S/I when the associated prime ideals of S/I are generated by disjoint sets of variables.

scholarly journals On asymptotic values of certain sets of attached prime ideals

1988 ◽  
Vol 30 (3) ◽  
pp. 293-300 ◽  
Author(s):  
A.-J. Taherizadeh
Keyword(s):  
Commutative Ring ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Primary Decomposition ◽  
Prime Divisors ◽  
Finitely Generated Module ◽  
Commutative Noetherian Ring ◽  
Asymptotic Values ◽  
Attached Prime ◽  
Positive Integers

In his paper [1], M. Brodmann showed that if M is a1 finitely generated module over the commutative Noetherian ring R (with identity) and a is an ideal of R then the sequence of sets {Ass(M/anM)}n∈ℕ and {Ass(an−1M/anM)}n∈ℕ (where ℕ denotes the set of positive integers) are eventually constant. Since then, the theory of asymptotic prime divisors has been studied extensively: in [5], Chapters 1 and 2], for example, various results concerning the eventual stable values of Ass(R/an;) and Ass(an−1/an), denoted by A*(a) and B*(a) respectively, are discussed. It is worth mentioning that the above mentioned results of Brodmann still hold if one assumes only that A is a commutative ring (with identity) and M is a Noetherian A-module, and AssA(M), in this situation, is regarded as the set of prime ideals belonging to the zero submodule of M for primary decomposition.

On the Finiteness Property of Generalized Local Cohomology Modules

2005 ◽  
Vol 12 (02) ◽  
pp. 293-300 ◽  
Author(s):  
K. Khashyarmanesh ◽  
M. Yassi
Keyword(s):  
Local Cohomology ◽  
Prime Ideals ◽  
Local Cohomology Module ◽  
Finiteness Property ◽  
Commutative Noetherian Ring ◽  
Local Cohomology Modules ◽  
Associated Prime Ideals ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules ◽  
Associated Prime

Let [Formula: see text] be an ideal of a commutative Noetherian ring R, and let M and N be finitely generated R-modules. Let [Formula: see text] be the [Formula: see text]-finiteness dimension of N. In this paper, among other things, we show that for each [Formula: see text], (i) the set of associated prime ideals of generalized local cohomology module [Formula: see text] is finite, and (ii) [Formula: see text] is [Formula: see text]-cofinite if and only if [Formula: see text] is so. Moreover, we show that whenever [Formula: see text] is a principal ideal, then [Formula: see text] is [Formula: see text]-cofinite for all n.

scholarly journals Mechanism Singularities Revisited From an Algebraic Viewpoint

2019 ◽  
Author(s):  
Zijia Li ◽  
Andreas Mueller
Keyword(s):  
Prime Ideal ◽  
Configuration Space ◽  
Smooth Point ◽  
Primary Ideal ◽  
Primary Decomposition ◽  
Algebraic Framework ◽  
Configuration Curve ◽  
Object Of Study ◽  
Kinematic Singularities ◽  
Associated Prime

Abstract It has become obvious that certain singular phenomena cannot be explained by a mere investigation of the configuration space, defined as the solution set of the loop closure equations. For example, it was observed that a particular 6R linkage, constructed by combination of two Goldberg 5R linkages, exhibits kinematic singularities at a smooth point in its configuration space. Such problems are addressed in this paper. To this end, an algebraic framework is used in which the constraints are formulated as polynomial equations using Study parameters. The algebraic object of study is the ideal generated by the constraint equations (the constraint ideal). Using basic tools from commutative algebra and algebraic geometry (primary decomposition, Hilbert’s Nullstellensatz), the special phenomenon is related to the fact that the constraint ideal is not a radical ideal. With a primary decomposition of the constraint ideal, the associated prime ideal of one primary ideal contains strictly into the associated prime ideal of another primary ideal which also gives the smooth configuration curve. This analysis is extended to shaky and kinematotropic linkages, for which examples are presented.

scholarly journals Ideals of Herzog–Northcott type

2011 ◽  
Vol 54 (1) ◽  
pp. 161-186 ◽  
Author(s):  
Liam O'Carroll ◽  
Francesc Planas-Vilanova
Keyword(s):  
Dimensional Space ◽  
Three Dimensional ◽  
Regular Sequence ◽  
Point Of View ◽  
Prime Ideals ◽  
Monomial Curves ◽  
Definition Of ◽  
Minimal Prime ◽  
Special Case ◽  
Three Dimensional Space

AbstractThis paper takes a new look at ideals generated by 2×2 minors of 2×3 matrices whose entries are powers of three elements not necessarily forming a regular sequence. A special case of this is the ideals determining monomial curves in three-dimensional space, which were studied by Herzog. In the broader context studied here, these ideals are identified as Northcott ideals in the sense of Vasconcelos, and so their liaison properties are displayed. It is shown that they are set-theoretically complete intersections, revisiting the work of Bresinsky and of Valla. Even when the three elements are taken to be variables in a polynomial ring in three variables over a field, this point of view gives a larger class of ideals than just the defining ideals of monomial curves. We then characterize when the ideals in this larger class are prime, we show that they are usually radical and, using the theory of multiplicities, we give upper bounds on the number of their minimal prime ideals, one of these primes being a uniquely determined prime ideal of definition of a monomial curve. Finally, we provide examples of characteristic-dependent minimal prime and primary structures for these ideals.

scholarly journals On two-sided artinian quotient rings

1972 ◽  
Vol 13 (2) ◽  
pp. 159-163 ◽  
Author(s):  
P. F. Smith
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Regular Element ◽  
Quotient Ring ◽  
Prime Ideals ◽  
Commutative Noetherian Ring ◽  
Zero Divisors ◽  
Quotient Rings

Djabali [1] has proved that, if R is a right and left noetherian ring with an identity and if the proper prime ideals of R are maximal, then R has a right and left artinian two-sided quotient ring. Robson [5, Theorem 2.11] and Small [6, Theorem 2.13] have proved independently that, if Ris a commutative noetherian ring, then Rhas an artinian quotient ring if and only if the prime ideals of Rthat belong to the zero ideal are all minimal. We shall generalise these results by proving theTheorem. Let R be a right and left noetherian ring with a regular element. Then R has a right and left artinian two-sided quotient ring if and only if each prime ideal of R consisting of zero–divisors is minimal.

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