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

Containment Problem for Quasi Star Configurations of Points in ℙ2

In this paper, the containment problem for the defining ideal of a special type of zero-dimensional subscheme of ℙ2, the so-called quasi star configuration, is investigated. Some sharp bounds for the resurgence of these types of ideals are given. As an application of this result, for every real number [Formula: see text], we construct an infinite family of homogeneous radical ideals of points in 𝕂[ℙ2] such that their resurgences lie in the interval [2−ε, 2). Moreover, the Castelnuovo-Mumford regularity of all ordinary powers of defining ideal of quasi star configurations are determined. In particular, it is shown that all of these ordinary powers have a linear resolution.

Program Deradikalisasi Radikalisme dan Terorisme Melalui Nilai-Nilai Luhur Pancasila

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. Keywords: Deradicalisation, Terrorism, Development, Radicalism

Fuzzy Semi-Maximal and Fuzzy Radical Ideals in MV-Algebras

In this paper, we introduce the notions of fuzzy semi-maximal ideals and fuzzy primary ideals of an [Formula: see text]-algebra and investigate some of their properties. Also, several characterizations of these fuzzy ideals are given. In addition, we show that [Formula: see text] is a fuzzy semi-maximal ideal of [Formula: see text] if and only if [Formula: see text] is a semi-simple [Formula: see text]-algebra and [Formula: see text] is a fuzzy primary ideal of [Formula: see text] if and only if [Formula: see text] is local [Formula: see text]-algebra. By using the notions of the maximal and normal fuzzy semi-maximal ideals, we show that under certain conditions a fuzzy semi-maximal ideal is two-valued and takes the values 0 and 1. The radical of a fuzzy ideal is defined as against the (maximal) radical of a fuzzy ideal and some of their properties are proved.

Commutative rings in which zero-components of essential primes are essential

For a prime ideal [Formula: see text] of a commutative ring [Formula: see text] with identity, we denote (as usual) by [Formula: see text] its zero-component; that is, the set of members of [Formula: see text] that are annihilated by nonmembers of [Formula: see text]. We study rings in which [Formula: see text] is an essential ideal, whenever [Formula: see text] is an essential prime ideal. We characterize them in terms of the lattices (which are, in fact, complete Heyting algebras) of their radical ideals. We prove that the classical ring of quotients of any ring of this kind is itself of this kind. We show that direct products of rings of this kind are themselves of this kind. We observe that the ring of real-valued continuous functions on a Tychonoff space is of this kind precisely when the underlying set of the space is infinite. Replacing [Formula: see text] with the pure part of [Formula: see text], we obtain a formally stronger variant which is still characterizable in terms of the lattices of radical ideals.

Open Set Lattices of Subspaces of Spectrum Spaces

We take a unified approach to study the open set lattices of various subspaces of the spectrum of a multiplicative lattice L. The main aim is to establish the order isomorphism between the open set lattice of the respective subspace and a sub-poset of L. The motivating result is the well known fact that the topology of the spectrum of a commutative ring R with identity is isomorphic to the lattice of all radical ideals of R. The main results are as follows: (i) for a given nonempty set S of prime elements of a multiplicative lattice L, we define the S-semiprime elements and prove that the open set lattice of the subspace S of Spec(L) is isomorphic to the lattice of all S-semiprime elements of L; (ii) if L is a continuous lattice, then the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m-semiprime elements of L; (iii) we define the pure elements, a generalization of the notion of pure ideals in a multiplicative lattice and prove that for certain types of multiplicative lattices, the sub-poset of pure elements of L is isomorphic to the open set lattice of the subspace Max(L) consisting of all maximal elements of L.