Radical Ideals Research Articles

Finite Convergence of Moment-SOS Relaxations with Nonreal Radical Ideals

Finite Convergence of Moment-SOS Relaxations with Nonreal Radical Ideals

Tilting and untilting for ideals in perfectoid rings

For a perfectoid ring R of characteristic 0 with tilt R♭\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{\\flat }$$\\end{document}, we introduce and study a tilting map (-)♭\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-)^{\\flat }$$\\end{document} from the set of p-adically closed ideals of R to the set of ideals of R♭\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{\\flat }$$\\end{document} and an untilting map (-)♯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-)^{\\sharp }$$\\end{document} from the set of radical ideals of R♭\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{\\flat }$$\\end{document} to the set of ideals of R. The untilting map (-)♯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-)^{\\sharp }$$\\end{document} is defined purely algebraically and generalizes the analytically defined untilting map on closed radical ideals of a perfectoid Tate ring of characteristic p introduced in the first author’s previous work. We prove that the two maps J↦J♭andI↦I♯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} J\\mapsto J^{\\flat }~\ ext {and}~I\\mapsto I^{\\sharp } \\end{aligned}$$\\end{document}define an inclusion-preserving bijection between the set of ideals J of R such that the quotient R/J is perfectoid and the set of p♭\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^{\\flat }$$\\end{document}-adically closed radical ideals of R♭\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{\\flat }$$\\end{document}, where p♭∈R♭\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^{\\flat }\\in R^{\\flat }$$\\end{document} corresponds to a compatible system of p-power roots of a unit multiple of p in R. Finally, we prove that the maps (-)♭\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-)^{\\flat }$$\\end{document} and (-)♯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(-)^{\\sharp }$$\\end{document} send (closed) prime ideals to prime ideals and thus define a homeomorphism between the subspace of Spec(R)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Spec}\\,}}(R)$$\\end{document} consisting of prime ideals p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {p}$$\\end{document} of R such that R/p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R/\\mathfrak {p}$$\\end{document} is perfectoid and the subspace of Spec(R♭)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Spec}\\,}}(R^{\\flat })$$\\end{document} consisting of p♭\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^{\\flat }$$\\end{document}-adically closed prime ideals of R♭\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$R^{\\flat }$$\\end{document}. In particular, we obtain a generalization and a new proof of the main result of the first author’s previous work which concerned prime ideals in perfectoid Tate rings.

Noetherian spectrum condition and the ring 𝒜[X

Let [Formula: see text] be an ascending chain of commutative rings with identity and let [Formula: see text] be the ring of polynomials with coefficient of degree [Formula: see text] in [Formula: see text] for each [Formula: see text] In the first part of this paper, we give a necessary and sufficient conditions for the ring [Formula: see text] to satisfy Noetherian spectrum (a ring [Formula: see text] is said to have Noetherian spectrum if [Formula: see text] satisfies the Ascending Chain Condition (ACC) on radical ideals). In particular, we extend classical results proved by Ohm and Pendleton in [J. Ohm and R. Pendleton, Rings with Noetherian spectrum, Duke Math. J. 35 (1968) 631–639] and by Hizem in [S. Hizem, Chain conditions in rings of the form A + XB[X] and A + XI[X], in Commutative Algebra and its Applications (de Gruyter, 2009), pp. 259–274]. The second part of this paper is motivated by the interesting results proved on uniformly [Formula: see text]-Noetherian rings by Kim et al. in [W. Qi, H. Kim, F. Wang, M. Chen and W. Zhao, Uniformly S-Noetherian rings (submitted)]. We introduce the concept of uniformly [Formula: see text]-Noetherian spectrum and study its properties. Let [Formula: see text] be a commutative ring and [Formula: see text] a multiplicative subset of [Formula: see text] We say that [Formula: see text] has uniformly[Formula: see text]-Noetherian spectrum if that there exists an [Formula: see text] such that for any ideal [Formula: see text] of [Formula: see text], [Formula: see text] for some finitely generated subideal [Formula: see text] of [Formula: see text] We give the Eakin–Nagata–Formanek theorem for the uniformly [Formula: see text]-Noetherian spectrum condition. We also study the uniformly [Formula: see text]-Noetherian uniformly properties on several ring constructions (Nagata’s idealization and amalgamated algebras).

Spitting at Jim Crow

Spitting at Jim Crow

On weakly radical ideals of non-commutative rings

For a commutative ring [Formula: see text] denote the Jacobson radical of the ring by [Formula: see text]. Khashan et al. introduced and studied J-ideals and weakly J-ideals for commutative rings with identity. In this paper, let [Formula: see text] be a non-commutative ring. We introduce weakly [Formula: see text]-ideals for a special radical [Formula: see text] and show that if [Formula: see text] is the Jacobson radical many of the results proved by Khashan et al. are also satisfied for non-commutative rings. The results for weakly J-ideals for non-commutative rings follow as special cases of a more general situation.

On radical ideals of non-commutative rings

Let [Formula: see text] denote the Jacobson radical of a commutative ring [Formula: see text]. In [H. A. Khashan and A. B. Bani-Ata, [Formula: see text]-ideals of commutative rings, Int. Electron. J. Algebra 29 (2021) 148–164], the notion of J-ideals was introduced. If [Formula: see text] denotes the prime radical of a commutative ring then in [U. Tekir, S. Koc and K. H. Oral, [Formula: see text]-Ideals of commutative rings, Filomat 31(10) (2017) 2933–2941], the notion of an [Formula: see text] ideal of a commutative ring was introduced and studied. In this note, we show that these results are special cases of a more general situation. We define [Formula: see text]-ideals for a special radical [Formula: see text] and prove that most of the results of the above-mentioned papers are satisfied for non-commutative rings as a special case.

New Examples of Radical Join-Meet Ideals

For non-distributive lattice L, it is not yet known classes of L with the property that the join-meet ideal of L is radical. In this paper, we give a partial answer to this problem.

The spectrum of a localic semiring

AbstractA number of spectrum constructions have been devised to extract topological spaces from algebraic data. Prominent examples include the Zariski spectrum of a commutative ring, the Stone spectrum of a bounded distributive lattice, the Gelfand spectrum of a commutative unital C*-algebra and the Hofmann–Lawson spectrum of a continuous frame.Inspired by the examples above, we define a spectrum for localic semirings. We use arguments in the symmetric monoidal category of suplattices to prove that, under conditions satisfied by the aforementioned examples, the spectrum can be constructed as the frame of overt weakly closed radical ideals and that it reduces to the usual constructions in those cases. Our proofs are constructive.Our approach actually gives ‘quantalic’ spectrum from which the more familiar localic spectrum can then be derived. For a discrete ring this yields the quantale of ideals and in general should contain additional ‘differential’ information about the semiring.

Deradicalization Through the Encouragement of Pancasila Values Education: Challenges for Indonesia and the International Community

Deradicalisation is every effort to neutralize radical ideals through interdisciplinary approaches, such as law, psychology, religion, economics, education, humanity and socio-culture for those who are influenced or exposed to radical and / or pro-violence. Implementation of Deradicalization Program (Development) can be done through Deradicalization in Prisons with the Target of terrorism prisoners who are in prison by identifying, Rehabilitation, Reeducation and Resosialisation. Deradicalisation has a goal, among other things, to restore the actors involved who have a radical understanding to return to a more moderate mindset. Deradicalisation of terrorism is very important because terrorism has become a serious problem for the international community because at any time it will endanger the national security for the country hence deradicalization program is needed as a formula of prevention and prevention of radical understanding like terrorism.

Extreme values of the resurgence for homogeneous ideals in polynomial rings

We show that two ostensibly different versions of the asymptotic resurgence introduced by E. Guardo, B. Harbourne and A. Van Tuyl in 2013 are the same. We also show that the resurgence and asymptotic resurgence attain their maximal values simultaneously, if at all, which we apply to a conjecture of E. Grifo. For radical ideals of points, we show that the resurgence and asymptotic resurgence attain their minimal values simultaneously. In addition, we introduce an integral closure version of the resurgence and relate it to the other versions of the resurgence. In closing we provide various examples and raise some related questions, and we finish with some remarks about computing the resurgence.

On Prime Ideals Related to an Ideal in a Commutative Ring

This article focuses on the notion of prime ideal related to an ideal of a commutative ring. Some of its characteristics are studied and the relations between ideals and prime ideals related to these ideals in a commutative ring are examined. Then some useful connections among them are obtained. Both similarities and differences between prime ideals related to an ideal and prime ideals are pointed out. Moreover a direct connection between prime ideals related to an ideal and radical ideals is obtained.

Finitary ideals of direct products in quantales

The notion of quantale, which designates a complete lattice equipped with an associative binary multiplication distributing over arbitrary joins, appears in various areasof mathematics-in quantaloid theory, in non classical logic as completion of the Lindebaum algebra, and in different representations of the spectrum of a C∗ algebra asmany-valued and non commutative topologies. To put it briefly, its importance is nolonger to be demonstrated. Quantales are ring-like structures in that they share withrings the common fact that while as rings are semi groups in the tensor category ofabelian groups, so quantales are semi groups in the tensor category of sup-lattices.In 2008 Anderson and Kintzinger [1] investigated the ideals, prime ideals, radical ideals, primary ideals, and maximal of a product ring R × S of two commutativenon non necceray unital rings R and S: Something resembling rings are quantales byanalogy with what is studied in ring, we begin an investigation on ideals of a productof two quantales. In this paper, given two quantales Q1 and Q2; not necessarily withidentity, we investigate the ideals, prime ideals, primary ideals, and maximal ideals ofthe quantale Q1 × Q2

Identifying limits of ideals of points in the case of projective space

We study the closure of the locus of radical ideals in the multigraded Hilbert scheme associated with a standard graded polynomial ring and the Hilbert function of a homogeneous coordinate ring of points in general position in projective space. In the case of projective plane, we give a sufficient condition for an ideal to be in the closure of the locus of radical ideals. For projective space of arbitrary dimension we present a necessary condition. The paper is motivated by the border apolarity lemma which connects such multigraded Hilbert schemes with the theory of ranks of polynomials.

Minimal depth of monomial ideals via associated radical ideals

Minimal depth of monomial ideals via associated radical ideals

On CP-frames

Let $mathcal{R}_c( L)$ be the pointfree version of $C_c(X)$, the subring of $C(X)$ whose elements have countable image. We shall call a frame $L $ a $CP$-frame if thering $mathcal{R}_c( L)$ is regular. % The main aim of this paper is to introduce $CP$-frames, that is $mathcal{R}_c( L)$ is a regular ring. We give some We give some characterizations of $CP$-frames and we show that $L$ is a $CP$-frame if and only if each prime ideal of $mathcal{R}_c ( L)$ is an intersection of maximal ideals if and only if every ideal of $mathcal{R}_c ( L)$ is a $z_c$-ideal. In particular, we prove that any $P$-frame is a $CP$-frame but not conversely, in general. In addition, we study some results about $CP$-frames like the relation between a $CP$-frame $L$ and ideals of closed quotients of $L$. Next, we characterize $CP$-frames as precisely those $L$ for which every prime ideal in the ring $mathcal{R}_c ( L)$ is a $z_c$-ideal. Finally, we show that this characterization still holds if prime ideals are replaced by essential ideals, radical ideals, convex ideals, or absolutely convex ideals.

Avoidance and absorbance

We study the two dual notions of prime avoidance and prime absorbance. We generalize the classical prime avoidance lemma to radical ideals. A number of new criteria are provided for an abstract ring to be compactly packed (C.P., every set of primes satisfies avoidance) or properly zipped (P.Z., every set of primes satisfies absorbance). Special consideration is given to the interaction with chain conditions and Noetherian-like properties. It is shown that a ring is both C.P. and P.Z. if and only if it has finite prime spectrum.

Commutative rings with invertible-radical factorization

In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties under homomorphic image and their transfer to various contexts of constructions such as direct product, trivial ring extension and amalgamated duplication of a ring along an ideal. Our results generate examples that enrich the current literature with new and original families of rings satisfying these properties.

Singular loci of reflection arrangements and the containment problem

This paper provides insights into the role of symmetry in studying polynomial functions vanishing to high order on an algebraic variety. The varieties we study are singular loci of hyperplane arrangements in projective space, with emphasis on arrangements arising from complex reflection groups. We provide minimal sets of equations for the radical ideals defining these singular loci and study containments between the ordinary and symbolic powers of these ideals. Our work ties together and generalizes results in Bauer et al. (Int Math Res Not IMRN 24:7459–7514, 2019), Dumnicki et al. (J Algebra 393:24–29, 2013), Harbourne and Seceleanu (J Pure Appl Algebra 219(4):1062–1072, 2015) and Malara and Szpond (J Pure Appl Algebra 222(8):2323–2329, 2018) under a unified approach.

Semistar ascending chain conditions over polynomial rings

Let D be an integral domain. G. Picozza associated to a stable semistar operation $$\star $$ on D, a semistar operation $$\star _1$$ on the polynomial ring D[X]. We defined the notion of semistar accr domain. We prove that D is $${\widetilde{\star }}$$ -Noetherian if and only D[X] is $$\star _1$$ -accr. On the other hand, we prove that D satisfies the ascending chain condition on radical quasi- $${\widetilde{\star }}$$ -ideals if and only if D[X] satisfies the ascending chain condition on radical quasi- $$\star _1$$ -ideals if and only if the Nagata ring of D with respect to the semistar operation $${\widetilde{\star }}$$ satisfies the ascending chain condition on radical ideals.

Pandangan dan Konsep Deradikalisasi Beragama Perspektif Nahdlatul Ulama (NU)


 Abstrak
 Maraknya radikalisme yang mengatasnamakan agama baik sebagai faham dan gerakan menjadi perhatian serius pemerintah Indonesia dalam membendungnya. Pemerintah Indonesia melalui Badan Nasional Penanggulangan Terorisme (BNPT) berusaha melakukan langkah-langkah strategis dalam mengkonter gerakan radikal tersebut, salah satunya dengan menggandeng ormas Islam Nahdlatul Ulama (NU). Nahdlatul Ulama sebagai organisasi kemasyarakatan yang berfaham moderat, di samping juga memiliki modal jaringan organisasi yang kuat dan luas yang dapat mencapai akar rumput sehingga diharapkan sangat strategis dalam menkonter penyebaran faham radikal dan ekstrem di Indonesia. Konsep deradikalisasi yang diterapkan Nahdlatul Ulama dilakukan dengan cara persuasif dengan mengedepankan pendekatan humanis dalam membangun dialog di kalangan internal masyarakat Islam. Misalnya, melalui gerakan dakwah para ulama dan kiai NU, melalui jejaring pesantren, melalui kajian-kajian ilmiah seperti bahtsul masail, media dakwah online NU, serta dengan membentengi lembaga pendidikan NU berbasis Aswaja (Ahlussunah wal Jamaah).
 Kata kunci: deradikalisasi, agama, dan Nahdlatul Ulama