Integral Closure of a Ring Whose Regular Ideals Are Finitely Generated

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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Filtrations, closure operations and prime divisors

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100065221 ◽

1988 ◽

Vol 104 (1) ◽

pp. 31-46 ◽

Cited By ~ 6

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.

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FAILURE OF CANCELLATION FOR QUARTIC AND HIGHER-DEGREE ORDERS

Journal of Algebra and Its Applications ◽

10.1142/s021949880200029x ◽

2002 ◽

Vol 01 (04) ◽

pp. 469-481 ◽

Cited By ~ 2

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.

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Integral closure of a marot ring whose regular ideals are finitely generated*

Communications in Algebra ◽

10.1080/00927879908826528 ◽

1999 ◽

Vol 27 (4) ◽

pp. 1783-1795 ◽

Cited By ~ 2

Author(s):

Gyu Whan Chang

Keyword(s):

Integral Closure ◽

Finitely Generated

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On the Stanley Depth of Powers of Monomial Ideals

Mathematics ◽

10.3390/math7070607 ◽

2019 ◽

Vol 7 (7) ◽

pp. 607 ◽

Cited By ~ 1

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.

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The asymptotic closure of an ideal relative to a module

Tamkang Journal of Mathematics ◽

10.5556/j.tkjm.32.2001.379 ◽

2001 ◽

Vol 32 (3) ◽

pp. 231-235

Author(s):

Sylvia M. Foster ◽

Johnny A. Johnson

Keyword(s):

Commutative Ring ◽

Integral Closure ◽

Finitely Generated ◽

Finitely Generated Module ◽

Noetherian Module ◽

The Ideal

In this paper we introduce the concept of the asymptotic closure of an ideal of a commutative ring $ R $ with identity relative to a unitary $ R $-module $ M $. This work extends results from P. Samuel, M. Nagata, J. W. Petro and Sharp, Tiras, and Yassi. Our objectives in this paper are to establish the cancellation law for the asymptotic completion of an ideal relative to a finitely generated module and show that the integral closure of an ideal relative to a Noetherian module $ M $ coincides with the asymptotic closure of the ideal relative to the Noetherian module $ M $.

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A Lipman’s type construction, glueings and complete integral closure

Nagoya Mathematical Journal ◽

10.1017/s0027763000001276 ◽

1989 ◽

Vol 113 ◽

pp. 99-119 ◽

Cited By ~ 5

Author(s):

Valentina Barucci

Keyword(s):

Jacobson Radical ◽

Integral Closure ◽

Prime Ideals ◽

Finitely Generated ◽

Blowing Up ◽

Complete Integral Closure ◽

Type Construction

Given a semilocal 1-dimensional Cohen-Macauly ring A, J. Lipman in [10] gives an algorithm to obtain the integral closure Ā of A, in terms of prime ideals of A. More precisely, he shows that there exists a sequence of rings A = A0 ⊂ A1 ⊂… ⊂ Ai ⊂…, where, for each i, i ≥ 0, Ai+1 is the ring obtained from Ai by “blowing-up” the Jacobson radical ℛ i of Ai+ i.e. Ai+l = ∪n(ℛin:ℛin). It turns out that ∪ {Ai;i≥0} = Ā (cf. [10, proof of Theorem 4.6]) and, if Ā is a finitely generated A-module, the sequence {Ai; i ≥ 0} is stationary for some m and Am = Ā, so that

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Automorphisms of finite order of nilpotent groups II

Studia Scientiarum Mathematicarum Hungarica ◽

10.1556/sscmath.51.2014.4.1297 ◽

2014 ◽

Vol 51 (4) ◽

pp. 547-555 ◽

Cited By ~ 1

Author(s):

B. Wehrfritz

Keyword(s):

Nilpotent Group ◽

Finite Order ◽

Finite Index ◽

Nilpotent Groups ◽

Finitely Generated

Let G be a nilpotent group with finite abelian ranks (e.g. let G be a finitely generated nilpotent group) and suppose φ is an automorphism of G of finite order m. If γ and ψ denote the associated maps of G given by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\gamma :g \mapsto g^{ - 1} \cdot g\phi and \psi :g \mapsto g \cdot g\phi \cdot g\phi ^2 \cdots \cdot \cdot g\phi ^{m - 1} for g \in G,$$ \end{document} then Gγ · kerγ and Gψ · ker ψ are both very large in that they contain subgroups of finite index in G.

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