Finitely generated prime ideals inH ? andD

1986 ◽  
Vol 191 (2) ◽  
pp. 297-302 ◽  
Author(s):  
Raymond Mortini
Keyword(s):  
Prime Ideals ◽  
Finitely Generated

scholarly journals Characterizations of m-Normal Nearlattices in terms of Principal n-Ideals

2013 ◽  
Vol 38 ◽  
pp. 49-59
Author(s):  
MS Raihan
Keyword(s):  
Prime Ideal ◽  
Prime Ideals ◽  
Single Element ◽  
Finitely Generated ◽  
Fixed Element ◽  
Minimal Prime

A convex subnearlattice of a nearlattice S containing a fixed element n?S is called an n-ideal. The n-ideal generated by a single element is called a principal n-ideal. The set of finitely generated principal n-ideals is denoted by Pn(S), which is a nearlattice. A distributive nearlattice S with 0 is called m-normal if its every prime ideal contains at most m number of minimal prime ideals. In this paper, we include several characterizations of those Pn(S) which form m-normal nearlattices. We also show that Pn(S) is m-normal if and only if for any m+1 distinct minimal prime n-ideals Po,P1,…., Pm of S, Po ? … ? Pm = S. DOI: http://dx.doi.org/10.3329/rujs.v38i0.16548 Rajshahi University J. of Sci. 38, 49-59 (2010)

Annihilator of local cohomology of homogeneous parts of a graded module

2020 ◽  
pp. 2150092
Author(s):  
Tran Do Minh Chau ◽  
Nguyen Thi Kieu Nga ◽  
Le Thanh Nhan
Keyword(s):  
Asymptotic Stability ◽  
Local Ring ◽  
Local Cohomology ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Graded Ring ◽  
Noetherian Local Ring ◽  
Graded Module

Let [Formula: see text] be a homogeneous graded ring, where [Formula: see text] is a Noetherian local ring. Let [Formula: see text] be a finitely generated graded [Formula: see text]-module. For [Formula: see text] set [Formula: see text]. Denote by [Formula: see text] the set of all prime ideals of [Formula: see text] containing [Formula: see text]. For [Formula: see text], let [Formula: see text] be the set of all [Formula: see text] such that [Formula: see text] In this paper, we prove that the sets [Formula: see text] and [Formula: see text] do not depend on [Formula: see text] for [Formula: see text]. We show that the annihilators [Formula: see text], [Formula: see text] are eventually stable, where [Formula: see text] for [Formula: see text]. As an application, we prove the asymptotic stability of some loci contained in the non-Cohen–Macaulay locus of [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.

scholarly journals Standard prime ideals and lying over for finite extensions of Noetherian algebras

1995 ◽  
Vol 37 (3) ◽  
pp. 311-326
Author(s):  
Günter Krause
Keyword(s):  
Prime Factor ◽  
Finite Sequence ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Associated Prime ◽  
Kirillov Dimension

Let k be a field, let R be a noetherian k-algebra of finite Gelfand-Kirillov dimension GK(R), and let M be a finitely generated right R-module. A standard prime factor series for M is a finite sequence of submodules 0 = N0 ⊂ N1 ⊂…⊂ Ni−1 ⊂ Ni ⊂.… ⊂ Nn = M, such that for each i the annihilator Pi = rR (Ni/Ni−1) is the unique associated prime of Ni/Ni−1 and GK(R/Pi)≤ GK(R/Pj) whenever i≤ j. The set of prime ideals arising from such a series is an invariant of M, called the set of standard primes St(M) of M. The concept, inspired by the notion of a standard affiliated series introduced by Lenagan and Warfield in [7], has been developed in [5], where it was shown that St(M) coincides with the set of all those prime ideals that are minimal over the annihilator of a nonzero submodule of M.

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.

scholarly journals The Logical Complexity of Finitely Generated Commutative Rings

2018 ◽  
Vol 2020 (1) ◽  
pp. 112-166 ◽  
Author(s):  
Matthias Aschenbrenner ◽  
Anatole Khélif ◽  
Eudes Naziazeno ◽  
Thomas Scanlon
Keyword(s):  
Commutative Ring ◽  
Commutative Rings ◽  
Prime Ideals ◽  
Zariski Topology ◽  
Dual Numbers ◽  
Finitely Generated ◽  
Nonstandard Model

AbstractWe characterize those finitely generated commutative rings which are (parametrically) bi-interpretable with arithmetic: a finitely generated commutative ring A is bi-interpretable with $(\mathbb{N},{+},{\times })$ if and only if the space of non-maximal prime ideals of A is nonempty and connected in the Zariski topology and the nilradical of A has a nontrivial annihilator in $\mathbb{Z}$. Notably, by constructing a nontrivial derivation on a nonstandard model of arithmetic we show that the ring of dual numbers over $\mathbb{Z}$ is not bi-interpretable with $\mathbb{N}$.

scholarly journals A Remark on Coherent Overrings

1978 ◽  
Vol 21 (3) ◽  
pp. 373-375 ◽  
Author(s):  
Ira J. Papick
Keyword(s):  
Commutative Ring ◽  
Integral Domain ◽  
General Theorem ◽  
Krull Dimension ◽  
Prime Ideals ◽  
Finitely Generated ◽  
Quotient Field ◽  
Valuation Rings ◽  
Finitely Presented

Throughout this note, let R be a (commutative integral) domain with quotient field K. A domain S satisfying R ⊆ S ⊆ K is called an overring of R, and by dimension of a ring we mean Krull dimension. Recall [1] that a commutative ring is said to be coherent if each finitely generated ideal is finitely presented.In [2], as a corollary of a more general theorem, Davis showed that if each overring of a domain R is Noetherian, then the dimension of R is at most 1. (This corollary is the converse of a version of the Krull-Akizuki Theorem [5, Theorem 93], and can also be proved directly by using the existence of valuation rings dominating finite chains of prime ideals [4, Corollary 16.6].) It is our purpose to prove that if R is Noetherian and each overring of R is coherent, then the dimension of £ is at most 1. We shall also indicate some related questions and examples.

scholarly journals Factorization for bounded analytic functions in the unit disk and an application to prime ideals

2006 ◽  
Vol 99 (2) ◽  
pp. 168 ◽  
Author(s):  
Raymond Mortini
Keyword(s):  
Unit Disk ◽  
Analytic Functions ◽  
Prime Ideals ◽  
Open Unit ◽  
Open Unit Disk ◽  
Finitely Generated ◽  
Bounded Analytic Functions

We prove several factorization theorems for bounded analytic functions in the open unit disk and present a very simple new proof of two conjectures of Frank Forelli and the author on the structure of finitely generated, respectively countably generated prime ideals in $H^{\infty}$.

scholarly journals A note on the countable union of prime submodules

2001 ◽  
Vol 27 (10) ◽  
pp. 641-643 ◽  
Author(s):  
M. R. Pournaki ◽  
M. Tousi
Keyword(s):  
Noetherian Ring ◽  
Countable Family ◽  
Prime Ideals ◽  
Countable Union ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Prime Submodules

LetMbe a finitely-generated module over a Noetherian ringR. Suppose𝔞is an ideal ofRand letN=𝔞Mand𝔟=Ann(M/N). If𝔟⫅J(R),Mis complete with respect to the𝔟-adic topology,{Pi}i≥1is a countable family of prime submodules ofM, andx∈M, thenx+N⫅∪i≥1Piimplies thatx+N⫅Pjfor somei≥1. This extends a theorem of Sharp and Vámos concerning prime ideals to prime submodules.

scholarly journals Ideals in the Wiener algebra W+

1989 ◽  
Vol 46 (2) ◽  
pp. 220-228 ◽  
Author(s):  
Raymond Mortini ◽  
Michael von Renteln
Keyword(s):  
Banach Algebra ◽  
Taylor Series ◽  
Unit Disc ◽  
Prime Ideals ◽  
Open Unit ◽  
Open Unit Disc ◽  
Finitely Generated ◽  
Wiener Algebra

AbstractLet W+ denote the Banach algebra of all absolutely convergent Taylor series in the open unit disc. We characterize the finitely generated closed and prime ideals in W+. Finally, we solve a problem of Rubel and McVoy by showing that W+ is not coherent.

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