On reduced rings with prime factors simple

We obtain a common generalization of the results by Wong and Birkenmeier-Kim-Park, respectively, which say that a reduced ring with unity is strongly (respectively, weakly) regular if and only if all of its prime homomorphic images are division rings (respectively, simple domains). Our arguments are different from those in the known proofs and are quite simple. They also give a characterization of weakly regular reduced rings without unity. This characterization implies in particular that the class of weakly regular reduced rings forms a radical class. However, even if a weakly regular reduced ring has no unity, its prime homomorphic images must be simple domains with unity. In the second part of the paper, we study reduced rings whose prime homomorphic images are simple domains (not necessarily with unity).

Algebraic Structure of Supernilpotent Radical Class Constructed from a Topology Thychonoff Space

A radical class of rings is called a supernilpotent radicals if it is hereditary and it contains the class for some positive integer In this paper, we start by exploring the concept of Tychonoff space to build a supernilpotent radical. Let be a Tychonoff space that does not contain any isolated point. The set of all continuous real-valued functions defined on is a prime essential ring. Finally, we can show that the class of rings is a supernilpotent radical class containing the matrix ring .

The lumpen in Marx’s works and its relevance for contemporary political struggle

The role of class struggle in historical materialism and Marx’s works is central in explaining political, social, and historical phenomena. While the two main classes, the proletariat and bourgeoisie, drive the progress of capitalist society, Marx also includes references to other classes such as the lumpen that are historically relevant in class struggle. This article puts forward that Marx’s understanding of the lumpen continually changes throughout his works and is integral, not just peripheral, to class struggle. By examining the chronology of Marx’s definition of the lumpen, I argue that Marx’s later works break from his earlier indictments of the lumpen as counterrevolutionary when he concludes in Capital that the lumpen are an exploited class due to their relation to productive labor in capitalism. This approach to understanding the lumpen in Marx’s works leads me to argue that the lumpen can be a possible revolutionary force in revolutionary class struggle. Connecting Marx’s works to contemporary times, I show that traditional lumpen ways of production and life are becoming more ubiquitous, due to recent political and economic trends, and therefore more important to incorporate into political movements. I will contend that current political discourse has worked to discount the political actions of the lumpen in countries such as the United States and the United Kingdom, while at the same time the lumpen class has considerably grown in such countries. I conclude that the political left must incorporate an understanding of the lumpen into its struggles if it is to seriously create a radical class–based politics in the near future.

The Democracy Development Machine

What forces hinder decolonization efforts on the neoliberal terrain? In the aftermath of a genocidal scorched earth campaign, Mayas in the town of San Pedro Necta encountered a formidable democracy-development machine designed to displace radical class politics into private market advancement and local, indigenous-led electoral politics. Sampedranos regarded neoliberal democracy and development not as empty, depoliticized forms or colonial impositions, but as hard-won victories that met immediate needs and echoed revolutionary and local struggles. This historical ethnography examines how these governmentalized spaces fell short, simultaneously enabling and disfiguring an ethnic resurgence that fractured in a dispiriting atmosphere of pessimism, self-interest, deception, and mistrust. These dynamics fueled authoritarian populism but also radical reimaginings of democracy and development from below. These findings shed new light on rural politics in Guatemala and across neoliberal and post-conflict settings.

Prophet of the Electric Age: Cultural Performativity as Nonviolent Revolution in the Lifework of Percy Bysshe Shelley

This article examines Percy Bysshe Shelley’s interest in contemporary possibilities of text dissemination in order to reconcile the normally opposing tendencies of gradualism (“slow reform”) and “violent” revolution in his life and writings. I offer a close reading of two parallel cultural events, both of which produced a national commotion that widely disseminated radical views—the “Peterloo” massacre at Manchester and Lord Eldon’s copyright rulings as Lord Chancellor. In both instances, the government’s attempts to control expression had the opposite effect due to the consequences of press coverage and political activism. In combination with nonviolent textual piracy, I argue that the circulation of the belief in poetry’s power concomitantly with the formation of a radical canon encouraged the latter’s circulation as propaganda, de facto establishing a common cultural heritage for the growing radical movement. Since Shelley’s writings became a fundamental component of the cultural glue that encouraged cohesion of the expanding radical class as soon as one decade after his premature death, I suggest that a reading of Shelley’s political strategies that moves beyond “ineffectualism” can highlight the continuing relevance of Shelley’s aesthetics and political thought.

On a type of distributivity of lattice ordered groups

Let m be an infinite cardinal. Inspired by a result of Sikorski on m-representability of Boolean algebras, we introduce the notion of r m-distributive lattice ordered group. We prove that the collection of all such lattice ordered groups is a radical class. Using the mentioned notion, we define and investigate a homogeneity condition for lattice ordered groups.

Torsion classes of generalized Boolean algebras

Torsion classes and radical classes of lattice ordered groups have been investigated in several papers. The notions of torsion class and of radical class of generalized Boolean algebras are defined analogously. We denote by T g and R g the collections of all torsion classes or of all radical classes of generalized Boolean algebras, respectively. Both T g and R g are partially ordered by the class-theoretical inclusion. We deal with the relation between these partially ordered collection; as a consequence, we obtain that T g is a Brouwerian lattice. W. C. Holland proved that each variety of lattice ordered groups is a torsion class. We show that an analogous result is valid for generalized Boolean algebras.

Relatively uniform convergences in archimedean lattice ordered groups

For an archimedean lattice ordered group G let G d and G∧ be the divisible hull or the Dedekind completion of G, respectively. Put G d∧ = X. Then X is a vector lattice. In the present paper we deal with the relations between the relatively uniform convergence on X and the relatively uniform convergence on G. We also consider the relations between the o-convergence and the relatively uniform convergence on G. For any nonempty class τ of lattice ordered groups we introduce the notion of τ-radical class; we apply this notion by investigating relative uniform convergences.

RADICALS WITH THE α-AMITSUR PROPERTY

A radical γ of rings is said to have the Amitsur property if for all rings A, γ(A[X]) = (γ(A[X]) ∩ A)[X]. Let Xα denote a set of indeterminates of cardinality α. We say that γ has the α-Amitsur property if for all rings A, γ(A[Xα]) = (γ(A[Xα]) ∩ A)[Xα]. We study properties of this type of radicals and show relationships with other known radicals for rings. A ring A is said to be an absolute γ-ring if A[x1,…, xn] ∈ γ, for any n ∈ ℕ. We show that A is an absolute 𝔾-ring for the Brown–McCoy radical 𝔾, if and only if A is in the radical class S determined by the unitary strongly prime rings. Moreover, A is an absolute nil ring if and only if A is an absolute J-ring, where J denotes the Jacobson radical.