Understanding how torsion theories are described and constructed is crucial to the study of torsion theory. Mutations of torsion theories have been studied as a method of constructing another torsion theory from a given one. We have already obtained how to mutate ordinary torsion theories into generalized torsion theories associated with a Serre subcategory. The paper investigates when the generalized torsion theories give ordinary torsion theories and when ordinary torsion theories provide each other.

Let R be a commutative unital ring and N be a submodule of an R-module M. We show that: 1) the semiprime radical is an invariant on submodules generated by the ascending chain of envelopes of a given submodule; 2) for rings that satisfy the radical formula, ⟨EM (0)⟩ is an idempotent radical leading to a torsion theory whose torsion class has nil R-modules and the torsion-free class has reduced R-modules; and, 3) Noetherian uniserial modules satisfy the semiprime radical formula and their semiprime radical is a nil module.

Suppose that [Formula: see text] is an injective structure of [Formula: see text]-Mod such that the class [Formula: see text] is closed for direct limits, then two modules in [Formula: see text] are isomorphic if there are maps in [Formula: see text] from each one of the modules into the other. Examples of module classes in such injective structures include (pure, coneat and RD-) injective modules. The idea of injective structures is slightly generalized to include [Formula: see text]-injective modules for a hereditary torsion theory [Formula: see text], as well. Thus providing a generalization of a classical result of Bumby’s and two recent ones by Macías-Díaz.

Abstract In this paper we attempt to generalize some classical results from Artinian/Noetherian ring theory to hereditary torsion theories. It is worthwhile noticing that the investigation of Artinian and Noetherian rings is an abundant source of findings, primarily related to the structure of rings and modules. By extending the concept of S-Noetherian rings, we explore totally Artinian and totally Noetherian rings. The primary goal of this note is to establish connections between totally Artinian and totally Noetherian rings, and to characterize the former, initially through localization at maximal ideals and later, as a result, by demonstrating that the ring A is totally Artinian if, and only if, it has an Artinian homomorphic image with a totally torsion kernel. The theory is further developed by addressing several open problems and presenting illustrative examples. The central aim of this paper is to offer examples of these types of rings by exploring their underlying structure.

This paper investigates a class of static spacetimes within higher-dimensional ([Formula: see text]) scalar–torsion theories featuring non-minimal derivative coupling and an active scalar potential. The spacetime structure is conformally related to a product space comprising a two-surface and a [Formula: see text]-dimensional submanifold. By analyzing the equations of motion, we demonstrate that the [Formula: see text]-dimensional submanifold must admit constant triplet structures, one of which corresponds to the torsion scalar. This condition allows the equations of motion to be reduced to a single highly nonlinear ordinary differential equation, referred to as the master equation. Our analysis reveals that the solutions to this model generically exhibit at least one naked singularity at the origin, ruling out the possibility of black hole or wormhole configurations. In the asymptotic region, the spacetimes converge to geometries with constant scalar curvature, which, in general, do not satisfy Einstein’s field equations. To further explore the behavior of the solutions, we employ a perturbative approach to linearize the master equation and construct first-order corrections. Additionally, we establish the local and global existence properties of the master equation and rigorously prove the non-existence of regular global solutions for [Formula: see text]. These findings provide valuable insights into the structure and limitations of the scalar–torsion theory under consideration.

This context outlines a basic view of the torsional behaviors through the application of various piezoelectric actuator and sensor layers in the laminated composite rectangular bar subjected to uniform thermal loading with considering the pyroelectric effect. The structure with the Saint-Venant torsion theory is modeled with the prismatic torsion formulations under applied mechanical torsion loading and then the results are presented to visualize the states of the torsional stiffness and the significance of the thermal field. In the current study, an analytical method for determining the pyroelectric effect and the control gain with the lamination angle dependence for the torsion angle is presented. The effect of one piezopolymer and two piezoceramic plates on the torsion angle considering the different number of layers and stacking sequence in the long laminated composite bar subjected to torsional torque are studied. The torsion angles are increased and decreased through the application of the electric control gain by the piezoelectric layers attached to the surface of the composite bar. The Abaqus software package is used to create a numerical model validation of the laminated composite bar without considering the pyroelectric effect. Results are presented to illustrate the pyroelectric effect and demonstrate its efficiency and accuracy.

Torsion theories play an important role in abelian categories and they have been widely studied in the last sixty years. In recent years, with the introduction of pretorsion theories, the definition has been extended to general (non-pointed) categories. Many examples have been investigated in several different contexts, such as topological spaces and topological groups, internal preorders, preordered groups, toposes, V-groups, crossed modules, etc. In this paper, we show that pretorsion theories naturally appear also in the “classical” framework, namely in abelian categories. We propose two ways of obtaining pretorsion theories starting from torsion theories. The first one uses “comparable” torsion theories, while the second one extends a torsion theory with a Serre subcategory. We also give a universal way of obtaining a torsion theory from a given pretorsion theory in additive categories. We conclude by providing several examples in module categories, internal groupoids, recollements and representation theory.

A point-free version of torsionfree classes and the Goldie torsion theory

Let A be a commutative ring and σ be a hereditary torsion theory in Mod − A . We introduce the concept of a totally σ–prime ideal, which is a generalization of prime and σ–prime ideals, and extend it to consider also submodules instead of ideals. The use of totally σ–prime ideals and submodules is closer to the arithmetic of elements than σ–prime, which originally were the natural generalization of prime number, with respect to a hereditary torsion theory or a multiplicatively closed subset S ⊆ A . In addition, they recover the original theory of prime ideals. Applications to rings and modules with chains conditions, and classical results of prime ideals and submodules, are studied in the context of totally σ–prime ideals and submodules as well as new characterizations of these, are studied.

A new torsion theory system for thin-walled structures, the Plate–Beam theory, was used to explore the lateral torsional buckling (LTB) performance of functionally graded material (FGM) thin-walled columns with open cross-section. The application details were presented using the T-section columns as the study carrier. The energy functional model was first determined, in the derivation process from which the warping constant, centroid and shear center were derived concomitantly. Afterward, the equilibrium differential equation model and the LTB critical load were formulated. The analytical process did not involve any sectorial coordinate operation. A numerical program was performed in parallel to evaluate the accuracy of theoretical equations. The section twist rotation and critical load error between the theoretical and numerical results were all lower than 2%. Especially, the warping displacement and warping normal stresses can be described with high precision. Based on the comprehensive theoretical analysis and numerical simulation, it was evident that the Plate–Beam theory was a reliable method for the FGM thin-walled member torsion performance study. The formulation process in this work provided a replicable direct reference for FGM members with other types of open sections.

On black holes in teleparallel torsion theories of gravity

In this paper, we study BKN rings, which are rings that satisfy the property Hom ⁡ R ( M , N ) ≠ 0 for all nonzero left (right) R -modules M and N . Our aim is to provide a comprehensive overview of BKN rings and their connections to other types of rings, such as retractable and completely coretractable rings, commutative rings, and their connections to injective R -modules and non-trivial torsion theories. We prove several results, including the left-right symmetry of the BKN property and its preservation under various ring operations. Furthermore, we explore the relationship between BKN rings and the tensor product of modules. Our study provides insight into the comprehensive understanding of BKN rings, providing information about the relation between BKN rings and the Tor, Ext, and ext functors.

A theory of deformation of layered composite beams is developed based on the application of the asymptotic splitting method to a spatial problem of elasticity theory. A system of four ordinary differential equations with constant coefficients for three unknown macrodisplacements functions and an unknown function of the twist angle of the beam cross section is obtained. The type and order of these equations depend on the number of the asymptotic approximation. The coefficients of the specified system are integral characteristics of auxiliary boundary value problems in the cross section of the beam. The theory presented contains a system of four coupled equations, i.e. in the general case, the processes of bending in two planes, tension-compression and torsion are coupled. The theory obtained includes as a special case the following theories: the classical Bernoulli-Euler beam theory; the Timoshenko beam theory; the Saint-Venant free torsion theory; the Vlasov theory of thin-walled beams of open cross section.

An accurate and efficient method based on the dynamic stiffness matrix for analyzing wave propagation in defective lattice structures

In this paper, we introduce and investigate Tor-tilting comodules and weak silting comodules, providing a generalization of the established tilting and silting theories in the comodule category. Our exploration yields a comprehensive understanding of their fundamental properties, including their associated torsion theories. We delve into not only the relations between Tor-tilting and weak silting comodules but also the connections among them and (co) tilting as well as (co) silting comodules. Additionally, we define Tor-tilting coflat comodules and Tor-tilting coflat dimensions. Finally, we explore Tor-tilting and weak silting objects in the morphism category, demonstrating their equivalence.

Homotopy torsion theories

We explore the existence and configurations of static and slowly rotating neutron stars (NSs) within a specific truncation of teleparallel scalar torsion theory. In this model, a scalar field ϕ is non-minimally coupled to the torsion scalar as ξTϕ 2, in the presence of the scalar potential V(ϕ) = -μ 2 ϕ 2/2 + λϕ 4/4. We establish the hydrostatic equilibrium equations for the static scenario and numerically solve them for both interior and exterior regions, employing appropriate boundary conditions near the center and at a distant location far away from the star's surface. Radial profiles of metric functions and the scalar field, alongside mass-radius diagrams, are plotted, utilizing four different realistic equations of state (EOS). Our results align closely with observational constraints from the GW170817 event, revealing a maximal mass of 2.37 M ⊙ achieved with the BSk21 EOS for a coupling parameter ξ = 0.25. Extending our analysis to encompass slow rotation, we establish the relationship between the star's moment of inertia and its mass. Furthermore, we explore future observations of NSs utilizing the redshift surface observable. Finally, we demonstrate the validity of the universality relation between the two forms of normalized moment of inertia within teleparallel scalar torsion theory with non minimal coupling.

This article presents a new theoretical method for the torsional-flexural vibration response of a thin-walled beam with closed cross sections under a uniformly distributed moving load. Based on Vlasov’s theory and restrained torsion theory, a comprehensive equation of motion for torsional flexural vibration is established by considering the effects of additional torque caused by the change of the shear center and the center of mass and the warpage coefficient. Using the Fourier transform, Laplace transform, and their inverse transformations, the torsional-flexural vibration response expression of thin-walled beams with closed cross-sections under a uniformly distributed moving load was derived. The results calculated by the analytical solutions in this study were compared with those from finite element method, demonstrating the reliability and superiority of the analytical solutions. Compared with the method that ignores the warpage coefficient, the lateral displacement and torsion angle of the beam calculated by the proposed theoretical method decreased by more than 12.00%. The effect of the cross-sectional properties on the natural frequency of the torsional-flexural vibration of thin-walled beams is analyzed, and the results show that the moment of inertia and mass per unit length have a considerable influence on the torsional-flexural frequency, which increases with the moment of inertia and decreases with an increase in the mass per unit length. The influences of the load magnitude, load velocity, and load eccentricity on the dynamic response were further investigated. According to the results, the lateral displacement of the beam increased significantly with increases in the load eccentricity and load magnitude. In practical engineering, the conclusions of this study can be used to limit the torsional-flexural vibration response of simple maglev support bridges to ensure the safety of vehicles and passengers.

Comodule theories in Grothendieck categories and relative Hopf objects

On coherent systems of subobjectswith application to torsion theories