Notion Of Radical Research Articles

Intuitionistic Fuzzy Prime Radical and Intuitionistic Fuzzy Primary Ideal Of $\Gamma$-Ring

In this paper, we introduced the notion of intuitionistic fuzzy prime radical of an intuitionistic fuzzy ideal in $\Gamma$-rings. We also characterise intuitionistic fuzzy primary ideal of $\Gamma$-rings. We also analyse homomorphic behaviour of intuitionistic fuzzy primary ideal and intuitionistic fuzzy prime radical of $\Gamma$-rings.

On prime and radical subacts of acts over semigroups

This study aims at giving some new results about the prime and radical subacts of any act over a semigroup S. The notion of prime radical is used to define a closure operator in the category of S-acts, some categorical properties of which are studied. Moreover, a lattice structure is introduced for the collection of radical subacts of an S-act. Also, two different types of Zariski spaces of prime varieties in the category of acts over a semigroup are presented. It is shown that a certain type of Zariski space of acts is isomorphic to a lattice of radical subacts.

The soft Jacobson radical of a commutative ring

In this paper, the notion of the soft Jacobson radical of a ring is defined. A relationship between the soft Jacobson radical of a ring and Jacobson semisimple ring is established. Some properties of this notion have been studied under homomorphism.

Projective Covers of Flat Contramodules

Abstract We show that a direct limit of projective contramodules (over a right linear topological ring) is projective if it has a projective cover. A similar result is obtained for $\infty $-strictly flat contramodules of projective dimension not exceeding $1$, using an argument based on the notion of the topological Jacobson radical. Covers and precovers of direct limits of more general classes of objects, both in abelian categories with exact and with nonexact direct limits, are also discussed, with an eye towards the Enochs conjecture about covers and direct limits, using locally split (mono)morphisms as the main technique. In particular, we offer a simple elementary proof of the Enochs conjecture for the left class of an $n$-tilting cotorsion pair in an abelian category with exact direct limits.

Some New Properties of Fuzzy Maximal Regular Open Sets

A proper nonempty open subset of a fuzzy topological space is said to be a fuzzy maximal regular open set , if any regular open set which contains is or . The purpose of this paper is to study some new fundamental properties of fuzzy maximal regular open sets. The decomposition theorems for a fuzzy maximal regular open set are investigated. Notion and basic properties of radical of fuzzy maximal regular open sets are established, such as the law of fuzzy radical closure. Some new properties and characterization theorems of fuzzy maximal regular open set are achieved.

The Kostrikin Radical and Similar Radicals of Lie Algebras

The existing notion of the Kostrikin radical as a radical in the Kurosh–Amitsur sense on classes of Mal’tsev algebras over rings with 1/6 is not completely justified. More precisely, to the fullest extent it is true for classes of Lie algebras over fields of characteristic zero and, as shown in the given paper, classes of algebraic Lie algebras of degree not greater than n over rings with 1/n! at all n ≥ 1. Similar conclusions are obtained in the paper also for the Jordan, regular, and extremal radicals constructed analogously to the Kostrikin radical.

Communitarianism, Multiculturalism and Liberalism

In the first part of this text, the author exposes the main features of the liberal or civic state, because both communitarians and multiculturalists tend to criticize that type of state. Their critique of the liberal state and the liberal self as an unencumbered self is “culturalist” by its character. However, it is an expression of conceptual confusion, i.e. of their incomprehension of an essential difference between two conceptual levels: one that belongs to the purely normative rights-justifying perspective and the other that refers to the ontological perspective. Consequently, both of them reject the central liberal thesis according to which the right is prior to the good.The author agrees with an assessment of Richard Robson that multiculturalism is only a form of communitarianism. Contrary to communitarians and multiculturalists, he additionally argues that collective rights are incompatible with the civic state in its pure form because there are structural differences between civic and specific minority rights.Further, the author attempts to show that communitarianism and multiculturalism are forms of postmodernism. Namely, brought to their ultimate logical consequences, the mentioned orientations can be connected to the postmodern notion of radical, irreducible difference.In the conclusive part of the text, he summarizes the common points of communitarianism and multiculturalism and emphasizes the importance of these contemporary theoretical tendencies.

$G$-Weights and $p$-Local Rank

Let $k$ be field of characteristic $p$, andlet $G$ be any finite group with splitting field $k$. Assume that $B$ is a $p$-block of $G$.In this paper, we introduce the notion of radical $B$-chain $C_{B}$, and we show that the $p$-local rank of $B$ is equals the length of $C_{B}$. Moreover, we prove that the vertex of a simple $kG$-module $S$ is radical if and only if it has the same vertex of the unique direct summand, up to isomorphism, of the Sylow permutationmodule whose radical quotient is isomorphic to $S$. Finally, we prove the vertices of certain direct summands of the Sylow permutation module are bounds for the vertices of simple $kG$-modules.

Prime congruences of additively idempotent semirings and a Nullstellensatz for tropical polynomials

A new definition of prime congruences in additively idempotent semirings is given using twisted products. This class turns out to exhibit some analogous properties to the prime ideals of commutative rings. In order to establish a good notion of radical congruences it is shown that the intersection of all primes of a semiring can be characterized by certain twisted power formulas. A complete description of prime congruences is given in the polynomial and Laurent polynomial semirings over the tropical semifield \({\mathbb {T}}\), the semifield \(\mathbb {Z}_{\mathrm{max}}\) and the two element semifield \({\mathbb {B}}\). The minimal primes of these semirings correspond to monomial orderings, and their intersection is the congruence that identifies polynomials that have the same Newton polytope. It is then shown that the radical of every finitely generated congruence in each of these cases is an intersection of prime congruences with quotients of Krull dimension 1. An improvement of a result from Bertram and Easton (Adv Math 308:36–82, 2017) is proven which can be regarded as a Nullstellensatz for tropical polynomials.

PARTIALLY ORDERED SETS OF NON-TRIVIAL NILPOTENT π-Subgroups

In this paper, we introduce a subposet $\mathcal{L}_{\pi}(G)$ of a poset $\mathcal{N}_{\pi}(G)$ of all non-trivial nilpotent $\pi$-subgroups of a finite group $G$. We examine basic properties of subgroups in $\mathcal{L}_{\pi}(G)$ which contain the notion of both radical $p$-subgroups and centric $p$-subgroups of $G$. It is shown that $\mathcal{L}_{\pi}(G)$ is homotopy equivalent to $\mathcal{N}_{\pi}(G)$. As examples, we investigate in detail the case where symmetric groups.

Prime Radical of Loops and Ω-Loops. I

In this paper, main properties of a commutator of two normal subloops of a loop are considered. The notion of a prime radical of loops is introduced and its characterization as a set of strongly Engel elements is given. Also an Ω-prime radical of Ω-loops is defined and its elementwise characterization is given.

Minimal prime state filters and state radicals on state pseudo $${ BL}$$ B L -algebras

The aim of this paper is the study of some classes of state filters of a state pseudo BL-algebra. The concepts of minimal prime state filter and of state hyperarchimedean pseudo BL-algebra are introduced and a characterization of a state hyperarchimedean pseudo BL-algebra is presented. Also, we define the notion of a state radical of a state filter of a state pseudo BL-algebra, we present a characterization of a state radical and some of its properties. The algebra of state radicals of a state pseudo BL-algebra is studied.

Torsion Radicals and Torsion Classes of Cyclically Ordered Groups

Abstract The notion of torsion radical of cyclically ordered groups is defined analogously as in the case of lattice ordered groups. We denote by T the collection of all torsion radicals of cyclically ordered groups. For τ1, τ2 ∈ T, we put τ1 ∈ τ2 if τ1(G) □ τ2(G) for each cyclically ordered group G. We show that T is a proper class; nevertheless, we apply for T the usual terminology of the theory of partially ordered sets. We prove that T is a complete completely distributive lattice. The analogous result fails to be valid for torsion radicals of lattice ordered groups. Further, we deal with products of torsion classes of cyclically ordered groups.

The dual notion of the prime radical of a module

In this article, we study the second radical of a module over an arbitrary ring R as the dual notion of the prime radical of a module. We give some properties of the second radical and determine the second radical of some modules. We define the notion of m⁎-system and describe the second radical of submodules in terms of m⁎-systems. We investigate when the second radical of a module M is equal to the socle of M. In particular, we give a characterization of the socle of a noetherian module over a ring R such that the ring R/P is right artinian for every right primitive ideal P by using the concept of second radical. We also give a characterization of right quasi-duo artinian rings by using the second radical of an injective module.

Study on the Structure of Γ-Semihypergroups

In this article, we study Γ-semihypergroups introduced recently. We introduce the hyper versions of Green's relations in Γ-semihypergroups and give some related characterizations. We introduce the n-prime Γ-hyperideal and n-semiprime Γ-hyperideal of a Γ-semihypergroup and show that for any integer n ≥ 2, n-prime Γ-hyperideals are a generalization of prime Γ-ideals. We also give the relationship between n-prime Γ-hyperideals and Γ-hyperideal extensions in Γ-semihypergroups. Then, we deal with prime ideal and introduce the notion of prime radical in a Γ-semihypergroup M, and also, we obtain some results and relations among the prime radicals of M and S, where S is the left operator semihypergroup of M.

Insurgent Planning: Situating Radical Planning in the Global South

This article revisits the notion of radical planning from the standpoint of the global South. Emerging struggles for citizenship in the global South, seasoned by the complexities of state—citizen relations within colonial and post-colonial regimes, offer an historicized view indispensable to counter-hegemonic planning practices. The article articulates the notion of insurgent planning as radical planning practices that respond to neoliberal specifics of dominance through inclusion — that is, inclusive governance. It characterizes the guiding principles for insurgent planning practices as counter-hegemonic, transgressive and imaginative. The article contributes to two current conversations within planning scholarship: on the implication of grassroots insurgent citizenship for planning, and on (de)colonization of planning theory.

Prime radicals of AO-groups

The notion of the prime radical for a directed group is considered. A series of results on commutants of AO-groups is obtained. Prime radicals of AO-groups are described.

Different shapes of modernism/neoclassicism: Case study of Dusan Radic’s creative output

This paper deals with musical neoclassicism in the context of modernism, and shows that some neoclassical features, usually considered to be reactionary and retrograde, could and should be understood as modernist. This kind of contextualization is especially important when Serbian postwar music is in question. Neoclassicism in Serbian music began in a specific ideological milieu. Paradoxically, and despite its "call for order" characteristics, it turned out to be a kind of modernistic phenomenon. On one hand it received all typical shapes of European modernism, while on the other it remains as a typical idiom of postwar Serbian musical production, blending in perfectly with the postmodern. These issues are argued and proved through analysis of some pieces written by Dusan Radic during the 1950?s. We see a kind of "disguised" modernism in the composer's subversion of the norms of national romanticism and socialist realism in the neoclassical choir piece Gungulice (1953), a modernism that is based on Freud's "reaction-formation" notion. Then, a more radical kind of modernism is followed in the piece The List for chamber ensemble (1952? 1954), arguing the notion of radical in the context of Serbian music, and coming to the conclusion that different interpretations of this piece as avant-garde, radical, or neoclassical/modernist can occur as a consequence of the listener's position. In the case of The Tower of Sculls we observe the same kind of "disguised" modernism, but now directly interwoven with proto-postmodern elements, coming from popular music practices, which is also the case in other pieces by Dusan Radic. The composer?s nonconformity, lack of pretentiousness, and inclination toward the simplicity of everyday things form his modernistic identity.

Radical approach in BCH‐algebras

We define the notion of radical in BCH‐algebra and investigate the structure of [X; k], a viewpoint of radical in BCH‐algebras.

A radical splitting theorem for bernstein algebras

We introduce the notion of radical in Bernstein algebras and prove a splitting theorem, that is an analog of a well-known statement in classical varieties of algebras. Note that in this situation Bernstein algebras are more similar to solvable Lie and Malcev algebras (see [4], [6]) than to associative, Jordan or Binary Lie ones.Throughout the paper all algebras and vector spaces are finite dimensional over an algebraically closed field k of characteristic 0.