scholarly journals ON THE ASSOCIATED PRIME IDEALS AND THE DEPTH OF POWERS OF SQUAREFREE PRINCIPAL BOREL IDEALS

2019 ◽  
pp. 224-244 ◽  
Author(s):  
Jurgen Herzog ◽  
Bahareh Lajmiri ◽  
Farhad Rahmati
Keyword(s):  
Prime Ideals ◽  
Associated Prime Ideals ◽  
Associated Prime ◽  
Borel Ideals

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.

On the Finiteness Results of the Generalized Local Cohomology Modules

2009 ◽  
Vol 16 (02) ◽  
pp. 325-332 ◽  
Author(s):  
Amir Mafi
Keyword(s):  
Positive Integer ◽  
Local Cohomology ◽  
Prime Ideals ◽  
Local Cohomology Module ◽  
Local Cohomology Modules ◽  
Associated Prime Ideals ◽  
Generalized Local Cohomology Modules ◽  
Cohomology Modules ◽  
Infinite Set ◽  
Associated Prime

Let 𝔞 be an ideal of a commutative Noetherian local ring R, and let M and N be two finitely generated R-modules. Let t be a positive integer. It is shown that if the support of the generalized local cohomology module [Formula: see text] is finite for all i < t, then the set of associated prime ideals of the generalized local cohomology module [Formula: see text] is finite. Also, if the support of the local cohomology module [Formula: see text] is finite for all i < t, then the set [Formula: see text] is finite. Moreover, we prove that gdepth (𝔞+ Ann (M),N) is the least integer t such that the support of the generalized local cohomology module [Formula: see text] is an infinite set.

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 the stable set of associated prime ideals of a monomial ideal

2012 ◽  
Vol 98 (3) ◽  
pp. 213-217 ◽  
Author(s):  
Shamila Bayati ◽  
Jürgen Herzog ◽  
Giancarlo Rinaldo
Keyword(s):  
Monomial Ideal ◽  
Prime Ideals ◽  
Stable Set ◽  
Associated Prime Ideals ◽  
Associated Prime

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.

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.

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