Associated Prime Ideals in Non-Noetherian Rings

1984 ◽  
Vol 36 (2) ◽  
pp. 344-360 ◽  
Author(s):  
Juana Iroz ◽  
David E. Rush
Keyword(s):  
Prime Ideal ◽  
Finite Subset ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Associated Primes ◽  
The Past ◽  
Associated Prime Ideals ◽  
Associated Prime

The theory of associated prime ideals is one of the most basic notions in the study of modules over commutative Noetherian rings. For modules over non-Noetherian rings however, the classical associated primes are not so useful and in fact do not exist for some modules M. In [4] [22] a prime ideal P of a ring R is said to be attached to an R-module M if for each finite subset I of P there exists m ∊ M such that I ⊆ annR(m) ⊆ P. In [4] the attached primes were compared to the associated primes and the results of [4], [22], [23], [24] show that the attached primes are a useful alternative in non-Noetherian rings to associated primes. Several other methods of associating a set of prime ideals to a module M over a non-Noetherian ring have proven very useful in the past. The most common of these is the set Assf(M) of weak Bourbaki primes of M [2, pp. 289-290].

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.

ASSOCIATED PRIMES OVER ORE EXTENSION RINGS

2004 ◽  
Vol 03 (02) ◽  
pp. 193-205 ◽  
Author(s):  
SCOTT ANNIN
Keyword(s):  
Recent Work ◽  
Polynomial Rings ◽  
Prime Ideals ◽  
Associated Primes ◽  
Ore Extension ◽  
Skew Polynomial Rings ◽  
Associated Prime Ideals ◽  
Polynomial Module ◽  
Skew Polynomial ◽  
Associated Prime

The study of the prime ideals in Ore extension rings R[x,σ,δ] has attracted a lot of attention in recent years and has proven to be a challenging undertaking ([5], [7], [12], et al.). The present article makes a contribution to this study for the associated prime ideals. More precisely, we aim to describe how the associated primes of an R-module MR behave under passage to the polynomial module M[x] over an Ore extension R[x,σ,δ]. If we impose natural σ-compatibility and δ-compatibility assumptions on the module MR (see Sec. 2 below), we can describe all associated primes of the R[x,σ,δ]-module M[x] in terms of the associated primes of MR in a very straightforward way. This result generalizes the author's recent work [1] on skew polynomial rings.

scholarly journals U–essential prime divisors and sequences over an ideal

1986 ◽  
Vol 103 ◽  
pp. 39-66 ◽  
Author(s):  
Daniel Katz ◽  
Louis J. Ratliff
Keyword(s):  
Prime Ideal ◽  
Noetherian Ring ◽  
Asymptotic Theory ◽  
Minimal Prime Ideal ◽  
Standard Theory ◽  
Noetherian Rings ◽  
Associated Primes ◽  
Prime Divisors ◽  
Minimal Prime

All rings in this paper are assumed to be commutative with identity, and they will generally also be Noetherian.In several recent papers the asymptotic theory of ideals in Noetherian rings has been introduced and developed. In this new theory the roles played in the standard theory by associated primes, R-sequences, classical grade, and Cohen-Macaulay rings are played by, respectively, asymptotic prime divisors, asymptotic sequences, asymptotic grade, and locally quasi-unmixed Noetherian rings. And up to the present time it has been shown that quite a few results from the standard theory have a valid analogue in the asymptotic theory, and a number of interesting and useful new results concerning the asymptotic prime divisors of an ideal in a Noetherian ring have also been proved. In fact the analogy between the two theories is so good that a very useful (but not completely valid) working guide is: results from the standard theory should have a valid analogue in the asymptotic theory. And, although asymptotic sequences are coarser than R-sequences (for example, they behave nicely when passing to R/z with z a minimal prime ideal in R), the converse of this working guide has also proved useful.

A NOTE ON COMPLETELY PRIME IDEALS OF ORE EXTENSIONS

2010 ◽  
Vol 20 (03) ◽  
pp. 457-463 ◽  
Author(s):  
V. K. BHAT
Keyword(s):  
Prime Ideal ◽  
Prime Ideals ◽  
Recent Past ◽  
Associated Prime Ideals ◽  
Ore Extensions ◽  
Active Research ◽  
Considerable Work ◽  
Minimal Prime ◽  
Associated Prime ◽  
Completely Prime Ideal

The study of prime ideals has been an area of active research. In recent past a considerable work has been done in this direction. Associated prime ideals and minimal prime ideals of certain types of Ore extensions have been characterized. In this paper a relation between completely prime ideals of a ring R and those of R[x; σ, δ] has been given; σ is an automorphisms of R and δ is a σ-derivation of R. It has been proved that if P is a completely prime ideal of R such that σ(P) = P and δ(P) ⊆ P, then P[x; σ, δ] is a completely prime ideal of R[x; σ, δ]. It has also been proved that this type of relation does not hold for strongly prime ideals.

scholarly journals ON THE ASSOCIATED PRIMES OF LOCAL COHOMOLOGY

2018 ◽  
Vol 237 ◽  
pp. 1-9 ◽  
Author(s):  
HAILONG DAO ◽  
PHAM HUNG QUY
Keyword(s):  
Local Cohomology ◽  
Positive Characteristic ◽  
Short Proof ◽  
Singular Locus ◽  
Prime Ideals ◽  
Associated Primes ◽  
Commutative Noetherian Ring ◽  
Regular Sequences ◽  
Associated Prime Ideals ◽  
Associated Prime

Let $R$ be a commutative Noetherian ring of prime characteristic $p$. In this paper, we give a short proof using filter regular sequences that the set of associated prime ideals of $H_{I}^{t}(R)$ is finite for any ideal $I$ and for any $t\geqslant 0$ when $R$ has finite $F$-representation type or finite singular locus. This extends a previous result by Takagi–Takahashi and gives affirmative answers for a problem of Huneke in many new classes of rings in positive characteristic. We also give a criterion about the singularities of $R$ (in any characteristic) to guarantee that the set $\operatorname{Ass}H_{I}^{2}(R)$ is always finite.

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].

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

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].)

Associated prime ideals over skew PBW extensions

2020 ◽  
Vol 48 (12) ◽  
pp. 5038-5055
Author(s):  
Arturo Niño ◽  
María Camila Ramírez ◽  
Armando Reyes
Keyword(s):  
Prime Ideals ◽  
Associated Prime Ideals ◽  
Skew Pbw Extensions ◽  
Associated Prime

Asymptotic Behavior of Integral Closures in Modules

2007 ◽  
Vol 14 (03) ◽  
pp. 505-514 ◽  
Author(s):  
R. Naghipour ◽  
P. Schenzel
Keyword(s):  
Asymptotic Behavior ◽  
Integral Closure ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Large N ◽  
Nagata Ring ◽  
Associated Prime Ideals ◽  
Integral Closures ◽  
Definition Of ◽  
Associated Prime

Let R be a commutative Noetherian Nagata ring, let M be a non-zero finitely generated R-module, and let I be an ideal of R such that height MI > 0. In this paper, there is a definition of the integral closure Na for any submodule N of M extending Rees' definition for the case of a domain. As the main results, it is shown that the operation N → Na on the set of submodules N of M is a semi-prime operation, and for any submodule N of M, the sequences Ass R M/(InN)a and Ass R (InM)a/(InN)a(n=1,2,…) of associated prime ideals are increasing and ultimately constant for large n.

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