Perfect Rings Research Articles

Oscillatory chiral flows in confined active fluids with obstacles

An active colloidal fluid comprised of self-propelled spinning particles injecting energy and angular momentum at the microscale demonstrates spontaneous collective states that range from flocks to coherent vortices. Despite their seeming simplicity, the emergent far-from-equilibrium behavior of these fluids remains poorly understood, presenting a challenge to the design and control of next-generation active materials. When confined in a ring, such so-called polar active fluids acquire chirality once the spontaneous flow chooses a direction. In a perfect ring, this chirality is indefinitely long-lived. Here, we combine experiments on self-propelled colloidal Quincke rollers and mesoscopic simulations of continuum Toner-Tu equations to explore how such chiral states can be controlled and manipulated by obstacles. For different obstacle geometries three dynamic steady states have been realized: long-lived chiral flow, an apolar state in which the flow breaks up into counter-rotating vortices and an unconventional collective state with flow having an oscillating chirality. The chirality reversal proceeds through the formation of intermittent vortex chains in the vicinity of an obstacle. We demonstrate that the frequency of collective states with oscillating chirality can be tuned by obstacle parameters. We vary obstacle shapes to design chiral states that are independent of initial conditions. Building on our findings, we realize a system with two triangular obstacles that force the active fluid towards a state with a density imbalance of active particles across the ring. Our results demonstrate how spontaneous polar active flows in combination with size and geometry of scatterers can be used to control dynamic patterns of polar active liquids for materials design.

On rings with envelopes and covers regarding to C3, D3 and flat modules

In this paper, by taking the class of all [Formula: see text] (or [Formula: see text]) right [Formula: see text]-modules for general envelopes and covers, we characterize a semisimple artinian ring (or a right perfect ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes) and a right [Formula: see text]-ring (or a right noetherian [Formula: see text]-ring) via [Formula: see text]-covers (or [Formula: see text]-envelopes). By using isosimple-projective preenvelope, we obtained that if [Formula: see text] is a semiperfect right FGF ring (or left coherent ring), then every isosimple right [Formula: see text]-module has a projective cover. Moreover, we also characterize semihereditary serial rings (respectively, hereditary artinian serial rings) in terms of epic flat (respectively, projective) envelopes.

Lifting modules with finite internal exchange property and direct sums of hollow modules

A module [Formula: see text] is said to be lifting if, for any submodule [Formula: see text] of [Formula: see text], there exists a decomposition [Formula: see text] such that [Formula: see text] and [Formula: see text] is a small submodule of [Formula: see text]. A lifting module is defined as a dual concept of the extending module. A module [Formula: see text] is said to have the finite internal exchange property if, for any direct summand [Formula: see text] of [Formula: see text] and any finite direct sum decomposition [Formula: see text], there exists a direct summand [Formula: see text] of [Formula: see text] [Formula: see text] such that [Formula: see text]. This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any [Formula: see text]-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.

IS INFINITE DIMENSIONAL

AbstractGiven a perfect valuation ring $R$ of characteristic $p$ that is complete with respect to a rank-1 nondiscrete valuation, we show that the ring $\mathbb{A}_{\inf }$ of Witt vectors of $R$ has infinite Krull dimension.

Max-projective modules

Weakening the notion of [Formula: see text]-projectivity, a right [Formula: see text]-module [Formula: see text] is called max-projective provided that each homomorphism [Formula: see text], where [Formula: see text] is any maximal right ideal, factors through the canonical projection [Formula: see text]. We study and investigate properties of max-projective modules. Several classes of rings whose injective modules are [Formula: see text]-projective (respectively, max-projective) are characterized. For a commutative Noetherian ring [Formula: see text], we prove that injective modules are [Formula: see text]-projective if and only if [Formula: see text], where [Formula: see text] is [Formula: see text] and [Formula: see text] is a small ring. If [Formula: see text] is right hereditary and right Noetherian then, injective right modules are max-projective if and only if [Formula: see text], where [Formula: see text] is a semisimple Artinian and [Formula: see text] is a right small ring. If [Formula: see text] is right hereditary then, injective right modules are max-projective if and only if each injective simple right module is projective. Over a right perfect ring max-projective modules are projective. We discuss the existence of non-perfect rings whose max-projective right modules are projective.

The dual Schröder-Bernstein problem for modules

We prove that for a right perfect ring R, then the following hold: if A, B are projective right R-modules and there exist epimorphisms and then A is isomorphic to B. A consequence is the following: if A and B are right R-modules with A, an epimorphic image of B and B, an epimorphic image of A, then the projective cover of A and the projective cover of B are isomorphic.

The Dual Baer Criterion for non-perfect rings

Abstract Baer’s Criterion for Injectivity is a useful tool of the theory of modules. Its dual version (DBC) is known to hold for all right perfect rings, but its validity for the non-right perfect ones is a complex problem (first formulated by C. Faith [Algebra. II. Ring Theory, Springer, Berlin, 1976]). Recently, it has turned out that there are two classes of non-right perfect rings: (1) those for which DBC fails in ZFC, and (2) those for which DBC is independent of ZFC. First examples of rings in the latter class were constructed in [J. Trlifaj, Faith’s problem on R-projectivity is undecidable, Proc. Amer. Math. Soc. 147 2019, 2, 497–504]; here, we show that this class contains all small semiartinian von Neumann regular rings with primitive factors artinian.

RINGS WITH THE DUAL OF THE ISOMORPHISM THEOREM

A ring R satisfies the dual of the isomorphism theorem if R/Ra≅ 1(a) for all elements a of R, where 1(a) denotes the left annihilator. We call these rings left morphic. Examples include all unit regular rings and certain left uniserial local rings. We show that every left morphic ring is right principally injective, and use this to characterize the left perfect, right and left morphic rings.

Application of New Formulas for the Spiral Arm Formation to Selected Galaxies with Special Patterns

Seven special galaxies with very unusual spiral patterns are precisely simulated and studied by recently proposed ROTASE hypothesis with derived mathematical formulas for the spiral arm formation of the galaxies, fully match the prediction of the model. MCG+00-04-051 shows the broken connection of the spiral arms with the ends of the galactic bar, such broken connection is due to the termination of the X-matter emission by the emission-active black hole located at the center of the galaxy. The weak but still visible arm section in the image clearly proves that the disconnected arms do not rotate ahead of the closest bar ends, they actually rotate behind of the other ends of the bar from which they were generated. The galaxy NGC 4548 has apparent broken connection of spiral arms with bar ends in a low-quality image, such broken connection gives false impression that the arms rotate ahead of the galactic bar ends. However, weak and visible arm in the better-quality image clearly proves that the arms rotate behind the bar ends, same as galaxy MCG+00-04-051. ESO325-28 shows a perfect spiral ring pattern. The galaxy NGC 7098 is made of two identical rings, each ring is made of a half inner ring and a half outer ring, the two rings cross each other twice with chain-link style, this is due to the X-matter emission behavior change with time following Gaussian formula. UGC12646 shows the same pattern as NGC 7098 with two rings crossing each other twice with chain-link style. The pattern image of UGC12646 clearly demonstrates that quality of the spiral arms decreases with their age only, not with their distance to the ends of the bar. NGC 1079 shows the most amazing artwork-like double ring pattern, each spiral arm has three arm crossings: it crosses other arm twice and crosses itself once. Galaxy SDSS J015701.50-001644.4 shows a unique two-spiral arm structure, each arm structure is made of a half inner ring and a whole outer ring; the two outer rings cross each other with chain-link style. The result also shows that the galaxy NGC 4548 has incomplete or destroyed double ring pattern. The possible formation and evolution sequence of the Hoag’s object is illustrated with images of real galaxies. Pitch angles of spiral arms can be easily calculated after successful simulation of spiral patterns. The new formulas will be a new member of mathematical spiral equation family and have potential applications in architectures, industrious designs, artwork creations and descriptions of natural objects.

Early Time Evolution of Turbulence in the Space Environment by Neutral Beam Injection

AbstractThe Space Measurement of A Rocket‐released Turbulence mission will demonstrate the production and evolution of turbulence in the space environment via a cascade of plasma physics processes. A sounding rocket will inject neutral barium atoms into the upper ionosphere at high speed across the magnetic field. The barium atoms photoionize, and the magnetic field confines them, leading to an ion ring distribution in velocity space that is unstable to electrostatic lower hybrid waves. Nonlinear induced scattering of lower hybrid waves produces electromagnetic magnetosonic and whistler waves that rapidly escape the source region and propagate into the radiation belts. In this paper, we present models and simulations of early time parts of this cascade until the lower hybrid waves are generated. We study the neutral atom injection, expansion, and ionization analytically and computationally. In a realistic scenario, the velocity distribution function is found to be nongyrotropic instead of a perfect ring in the plane perpendicular to the magnetic field. Using these results, we examine a representative region of the barium cloud and perform particle‐in‐cell simulations of the excitation of the lower hybrid waves. Of particular interest is the energy extracted from the kinetic energy of the barium ions, as this will ultimately affect the amplitude of the electromagnetic waves in the radiation belts. We find that for velocity distribution functions that deviate from an ideal ring distribution, the energy extracted from the fast barium ions increases.

Some remarks on locally perfect rings

In this paper, the concept of almost projective covers is introduced. Using this concept, locally perfect rings are characterized. Also, the concepts of [Formula: see text]-almost projective modules, [Formula: see text]-almost projective covers, and [Formula: see text]-locally perfect rings are introduced. Then [Formula: see text]-locally perfect rings are characterized in terms of [Formula: see text]-almost projective modules and [Formula: see text]-almost projective covers, respectively.

Electrical control of spins and giant g-factors in ring-like coupled quantum dots

Emerging theoretical concepts for quantum technologies have driven a continuous search for structures where a quantum state, such as spin, can be manipulated efficiently. Central to many concepts is the ability to control a system by electric and magnetic fields, relying on strong spin-orbit interaction and a large g-factor. Here, we present a mechanism for spin and orbital manipulation using small electric and magnetic fields. By hybridizing specific quantum dot states at two points inside InAs nanowires, nearly perfect quantum rings form. Large and highly anisotropic effective g-factors are observed, explained by a strong orbital contribution. Importantly, we find that the orbital contributions can be efficiently quenched by simply detuning the individual quantum dot levels with an electric field. In this way, we demonstrate not only control of the effective g-factor from 80 to almost 0 for the same charge state, but also electrostatic change of the ground state spin.

Rings whose injective hulls are dual square free

A module M is called dual-square-free (DSF) if M has no proper submodules A and B with and The class of DSF-modules is closed under direct summands and homomorphic images, and a module M is distributive iff every submodule of M is dual-square-free. In this article we consider certain classes of rings R over which is a DSF-module. For example, if R is a right hereditary ring such that is DSF, then R is right noetherian, right distributive, every right ideal of R is two-sided, and every subfactor of RR is quasi-continuous. Also, if R is semilocal and is a DSF-module with small radical, then R is basic, semiperfect and right self-injective. As an immediate application, if R is a right perfect ring such that is DSF as a right R-module then R is right Artinian. If in addition we assume that either or is a DSF-module then the ring R becomes quasi-Frobenius.

Weakly dual automorphism invariant and superfluous cotightness

The aim of the present paper is to introduce and study the dual concepts of weakly automorphism invariant modules and essential tightness. These notions are non-trivial generalizations of both weakly projectivity, dual automorphism invariant property and cotightness. We obtain certain relations between weakly projective modules, weakly dual automorphism invariant modules and superfluous cotight modules. It is proved that: (1) for right perfect rings, every module is a direct summand of a weakly dual automorphism invariant module and (2) weakly dual automorphism invariant modules are precisely superfluous cotight modules.

On the p-typical de Rham–Witt complex over W(k)

Hesselholt and Madsen in [7] define and study the (absolute, p-typical) de Rham-Witt complex in mixed characteristic, where p is an odd prime. They give as an example an elementary algebraic description of the de Rham-Witt complex over Z_(p). The main goal of this paper is to construct, for k a perfect ring of characteristic p > 2, a Witt complex over A = W(k) with an algebraic description which is completely analogous to Hesselholt and Madsen's description for Z_(p). Our Witt complex is not isomorphic to the de Rham-Witt complex; instead we prove that, in each level, the de Rham-Witt complex over W(k) surjects onto our Witt complex, and that the kernel consists of all elements which are divisible by arbitrarily high powers of p. We deduce an explicit description of the de Rham-Witt complex over W(k). We also deduce results concerning the de Rham-Witt complex over certain perfectoid rings.

Envelopes, covers and semidualizing modules

Given an [Formula: see text]-module [Formula: see text] and a class of [Formula: see text]-modules [Formula: see text] over a commutative ring [Formula: see text], we investigate the relationship between the existence of [Formula: see text]-envelopes (respectively, [Formula: see text]-covers) and the existence of [Formula: see text]-envelopes or [Formula: see text]-envelopes (respectively, [Formula: see text]-covers or [Formula: see text]-covers) of modules. As a consequence, we characterize coherent rings, Noetherian rings, perfect rings and Artinian rings in terms of envelopes and covers by [Formula: see text]-projective, [Formula: see text]-flat, [Formula: see text]-injective and [Formula: see text]-[Formula: see text]-injective modules, where [Formula: see text] is a semidualizing [Formula: see text]-module.

S-almost perfect commutative rings

Given a multiplicative subset S in a commutative ring R, we consider S-weakly cotorsion and S-strongly flat R-modules, and show that all R-modules have S-strongly flat covers if and only if all flat R-modules are S-strongly flat. These equivalent conditions hold if and only if the localization RS is a perfect ring and, for every element s∈S, the quotient ring R/sR is a perfect ring, too. The multiplicative subset S⊂R is allowed to contain zero-divisors.

On image summand coinvariant modules and kernel summand invariant modules

In this paper we introduce the concept of im-summand coinvariance and im-small coinvariance; that is, a module $M$ over a right perfect ring is said to be im-summand (im-small) coinvariant if, for any endomorphism $\varphi$ of $P$ such that ${\rm Im} \varphi$ is a direct summand (a small submodule) of $P$, $\varphi (\ker \nu) \subseteq \ker \nu$, where $(P, \nu)$ is the projective cover of $M$. We first give some fundamental properties of im-summand coinvariant modules and im-small coinvariant modules, and we prove that, for modules $M$ and $N$ over a right perfect ring such that $N$ is a small epimorphic image of $M$, $M$ is $N$-im-summand coinvariant if and only if $M$ is (im-coclosed) $N$-projective. Moreover, we introduce ker-summand invariance and ker-essential invariance as the dual concept of im-summand coinvariance and im-small coinvariance, respectively, and show that, for modules $M$ and $N$ such that $N$ is isomorphic to an essential submodule of $M$, $M$ is $N$-ker-summand invariant if and only if $M$ is (ker-closed) $N$-injective.

Propagation of partially coherent truncated polymorphic beams

The recently introduced concept of coherent polymorphic beam (PB), which is focused into a 2D light curve of arbitrary form with independently prescribed phase along it, is a fruitful generalization of the perfect ring vortex and opens up new perspectives in all-optical particle manipulation and light material processing. Its application for optical transport of micro/nano-particles has been experimentally demonstrated. However, the propagation of the PB has not been studied yet. In this Letter, we derive analytical expressions for the propagation of the truncated PB and its partially coherent counter-part through the first-order optical systems, in particular, the rotationally symmetric and twisting systems described by the fractional Fourier and Gyrator transforms, respectively. These expressions clarify the light-curve formation from a truncated PB and can be easily applied for the numerical simulation of the partially coherent PB propagation

Edge deformation effects on sensitivity and selectivity performance of graphene quantum ring gas sensor

To explore the edge deformation effects on the performance of a graphene quantum ring gas sensor, we considered an armchair hexagonal graphene ring connected to two semi-infinite armchair graphene nanoribbons. We calculated the current through the graphene ring upon exposure to CO, NO, CO2, and NH3 gas molecules. It is shown that the behavior of current vs bias voltage depends on the inner and outer radii, and it is independent of the graphene quantum ring width. The effect of temperature resulting from nonequilibrium Green's function (Fermi–Dirac distribution) related to leads has been investigated. The substantial finding is that the current value remains unchanged up to room temperature at a perfect graphene quantum ring and, indeed, the sensor performance is unrelated to temperature. In a deformed graphene quantum ring, the influence of temperature on sensor performance is insignificant so that it is ineffective. Furthermore, the deformation in the edges can be accidental in the formation process, which we have simulated by randomly removing the atoms of the edges, or deformation can be manual, which was simulated by removing successive edge atoms from the ring sides. In the presence of edge deformation at a constant voltage, the difference between current values related to adsorption of NO, CO, and NH3 gas molecules and the pristine ring increases. In fact, the edge deformation improves the selectivity and the sensitivity of the graphene ring gas sensor. Single vacancy and double vacancies decrease the graphene ring sensor's performance. This underlines the importance of precision in the fabrication of nonedge parts of a graphene ring, although edge deformation is worthwhile in the improvement of the gas sensor.