Filtrations, closure operations and prime divisors

1988 ◽  
Vol 104 (1) ◽  
pp. 31-46 ◽  
Author(s):  
J. S. Okon ◽  
L. J. Ratliff
Keyword(s):  
Integral Closure ◽  
Finitely Generated ◽  
Prime Divisors ◽  
Closure Operation ◽  
Closed Set ◽  
Closure Operations

AbstractLet ƒ = {In}n ≽ 0 be a filtration on a ring R, let(In)w = {x ε R; x satisfies an equation xk + i1xk − 1 + … + ik = 0, where ij ε Inj} be the weak integral closure of In and let ƒw = {(In)w}n ≽ 0. Then it is shown that ƒ ↦ ƒw is a closure operation on the set of all filtrations ƒ of R, and if R is Noetherian, then ƒw is a semi-prime operation that satisfies the cancellation law: if ƒh ≤ (gh)w and Rad (ƒ) ⊆ Rad (h), then ƒw ≤ gw. These results are then used to show that if R and ƒ are Noetherian, then the sets Ass (R/(In)w) are equal for all large n. Then these results are abstracted, and it is shown that if I ↦ Ix is a closure (resp.. semi-prime, prime) operation on the set of ideals I of R, then ƒ ↦ ƒx = {(In)x}n ≤ 0 is a closure (resp., semi-prime, prime) operation on the set of filtrations ƒ of R. In particular, if Δ is a multiplicatively closed set of finitely generated non-zero ideals of R and (In)Δ = ∪KεΔ(In, K: K), then ƒ ↦ ƒΔ is a semi-prime operation that satisfies a cancellation law, and if R and ƒ are Noetherian, then the sets Ass (R/(In)Δ) are quite well behaved.

scholarly journals Integral Closure of a Ring Whose Regular Ideals Are Finitely Generated

2002 ◽  
Vol 251 (2) ◽  
pp. 529-537 ◽  
Author(s):  
Gyu Whan Chang ◽  
Byung Gyun Kang
Keyword(s):  
Integral Closure ◽  
Finitely Generated

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 Homomorphic Image and Inverse Image of Weak Closure Operations on Ideals of BCK-Algebras

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 567 ◽  
Author(s):  
Hashem Bordbar ◽  
Young Bae Jun ◽  
Seok-Zun Song
Keyword(s):  
Homomorphic Image ◽  
Inverse Image ◽  
Closure Operation ◽  
Weak Closure ◽  
Closure Operations ◽  
Epimorphic Image

We introduce the notions of meet, semi-prime, and prime weak closure operations. Using homomorphism of BCK-algebras φ : X → Y , we show that every epimorphic image of a non-zeromeet element is also non-zeromeet and, for mapping c l Y : I ( Y ) → I ( Y ) , we define a map c l Y ← on I ( X ) by A ↦ φ − 1 ( φ ( A ) c l Y ) . We prove that, if “ c l Y ” is a weak closure operation (respectively, semi-prime and meet) on I ( Y ) , then so is “ c l Y ← ” on I ( X ) . In addition, for mapping c l X : I ( X ) → I ( X ) , we define a map c l X → on I ( Y ) as follows: B ↦ φ ( φ − 1 ( B ) c l X ) . We show that, if “ c l X ” is a weak closure operation (respectively, semi-prime and meet) on I ( X ) , then so is “ c l X → ” on I ( Y ) .

FAILURE OF CANCELLATION FOR QUARTIC AND HIGHER-DEGREE ORDERS

2002 ◽  
Vol 01 (04) ◽  
pp. 469-481 ◽  
Author(s):  
RYAN KARR
Keyword(s):  
Mild Condition ◽  
Residue Field ◽  
Unit Group ◽  
Field Extension ◽  
Integral Closure ◽  
Principal Ideal Domain ◽  
Finitely Generated ◽  
Quotient Field ◽  
Almost All ◽  
Finitely Generated Modules

Let D be a principal ideal domain with quotient field F and suppose every residue field of D is finite. Let K be a finite separable field extension of F of degree at least 4 and let [Formula: see text] denote the integral closure of D in K. Let [Formula: see text] where f ∈ D is a nonzero nonunit. In this paper we show, assuming a mild condition on f, that cancellation of finitely generated modules fails for R, that is, there exist finitely generated R-modules L, M, and N such that L ⊕ M ≅ L ⊕ N and yet M ≇ N. In case the unit group of D is finite, we show that cancellation fails for almost all rings of the form [Formula: see text], where p ∈ D is prime.

scholarly journals The asymptotic and integral closure operations in multiplicative lattice modules

2005 ◽  
Vol 36 (4) ◽  
pp. 345-358
Author(s):  
Sylvia M. Foster ◽  
Johnny A. Johnson
Keyword(s):  
Chain Condition ◽  
Integral Closure ◽  
Multiplicative Lattice ◽  
Closure Operations ◽  
Ascending Chain Condition

This paper is primarily concerned with the integral and asymptotic closure operations on a multiplicative lattice relative to the greatest element of a lattice module having the ascending chain condition. We show that a cancellation law holds for the asymptotic closure of elements of the multiplicative lattice and we ultimately show, by means of multiplicative filtrations and filtration transforms, that the asymptotic closure of an element in a multiplicative lattice relative to the greatest element of a lattice module, coincides with its integral closure relative to this element in the lattice module.

scholarly journals On the Stanley Depth of Powers of Monomial Ideals

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 607 ◽  
Author(s):  
S. A. Seyed Fakhari
Keyword(s):  
Polynomial Ring ◽  
Upper Bound ◽  
Integral Closure ◽  
Monomial Ideals ◽  
The Other ◽  
Finitely Generated ◽  
Stanley Depth ◽  
Survey Article ◽  
Symbolic Powers ◽  
Other Hand

In 1982, Stanley predicted a combinatorial upper bound for the depth of any finitely generated multigraded module over a polynomial ring. The predicted invariant is now called the Stanley depth. Duval et al. found a counterexample for Stanley’s conjecture, and their counterexample is a quotient of squarefree monomial ideals. On the other hand, there is evidence showing that Stanley’s inequality can be true for high powers of monomial ideals. In this survey article, we collect the recent results in this direction. More precisely, we investigate the Stanley depth of powers, integral closure of powers, and symbolic powers of monomial ideals.

A note on prime divisors of polynomials P(Tk); k ≥ 1

2019 ◽  
Vol 69 (1) ◽  
pp. 213-222
Author(s):  
François Legrand
Keyword(s):  
Positive Integer ◽  
Number Field ◽  
Residue Field ◽  
Sufficient Conditions ◽  
Integral Closure ◽  
Prime Ideals ◽  
Prime Divisors ◽  
Reduction Modulo

Abstract Let F be a number field, OF the integral closure of ℤ in F, and P(T) ∈ OF[T] a monic separable polynomial such that P(0) ≠ 0 and P(1) ≠ 0. We give precise sufficient conditions on a given positive integer k for the following condition to hold: there exist infinitely many non-zero prime ideals 𝓟 of OF such that the reduction modulo 𝓟 of P(T) has a root in the residue field OF/𝓟, but the reduction modulo 𝓟 of P(Tk) has no root in OF/𝓟. This makes a result from a previous paper (motivated by a problem in field arithmetic) asserting that there exist (infinitely many) such integers k more precise.

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