Perfect Rings Research Articles

Tunable Multimode Plasmonic Filter Based on Side-Coupled Ring-Groove Joint Resonator

A tunable multimode plasmonic filter is proposed by using a side-coupled ring-groove joint resonator. In addition to the integer resonance modes of the perfect ring resonator (RR), extra non-integer resonance modes are excited by adding a groove on the RR. According to the simulations with finite difference time domain method, it is investigated that two integer and two non-integer modes are obtained. When the groove is placed on the antinodes of the magnetic fields, the effective resonance lengths for the corresponding modes will be changed by the groove. In this case, one can linearly manipulate the wavelengths by changing the length of the groove. On the contrary, the wavelengths of the specific modes will be always fixed when the groove locates at the nodes of the modes. Compared to those perfect structured Fabry-Perot (FP) resonators, this compact device can provide more channels and manipulate the wavelengths flexibly. Therefore, it may find widely applications in the on-chip optical signal processing area.

Module Decompositions Using Pairwise Comaximal Ideals

In this article, we show that for a given set of pairwise comaximal ideals {Xi}i∈I in a ring R with unity and any right R-module M with generating set Y and , M = ⊕i∈IC(Xi) if and only if for every y ∈ Y there exists a nonempty finite subset J ⊆ I and positive integers kj such that . We investigate this decomposition for a general class of modules. Our main theorem can be applied to a large class of rings including semilocal rings R with the Jacobson radical of R equal to the prime radical of R, left (or right) perfect rings, piecewise prime rings, and rings with ascending chain condition (ACC) on ideals and satisfying the right AR property on ideals. This decomposition generalizes the decomposition of a torsion abelian group into a direct sum of its p-components. We also develop a torsion theory associated with sets of pairwise comaximal ideals.

Rings whose cyclic modules are pure-injective or pure-projective

A famous theorem of algebra due to Osofsky states that “if every cyclic left R-module is injective, then R is semisimple”. Therefore, a natural question of this sort is: “What is the class of rings R for which every cyclic left R-module is pure-injective or pure-projective?” The goal of this paper is to answer this question. For instance, we show that if every cyclic left R-module is pure-injective, then R is a left perfect ring. As a consequence, a commutative coherent ring R is Artinian if and only if every cyclic R-module is pure-injective. Also, a commutative ring R is pure-semisimple (i.e., every R-module is pure-injective) if and only if all cyclic R-modules and all indecomposable R-modules are pure-injective. We obtain some generalizations of Osofsky's theorem in the cases R is semiprimitive or commutative coherent or a commutative semiprime Goldie ring. Finally, we show that a ring R is left Noetherian if and only if every cyclic left R-module is pure-projective. As a corollary of this result we obtain: if every cyclic left R-module is pure-injective and pure-projective, then R is a left Artinian ring. The converse is also true when R is commutative.

Weakly injective dimension and Almost perfect rings

In this paper, we study the weak-injective dimension and we characterize the global weak-injective dimension of rings. After we study the transfer of the global weak-injective dimension in some known ring construction. Finely westudy the transfer of almost perfect property in pullback and D+M constructions.

Restricted Perfect Group Rings

A ring R is a restricted right perfect ring if every proper homomorphic image of R is right perfect. A complete characterization of restricted right perfect group rings RG has been obtained when the f.c. center of the group G is nontrivial. The f.c. center of a group G is the set of all elements of G that have finitely many conjugates in G.

DC-projective dimensions, Foxby equivalence and SDC-projective modules

This is a study of Ding projective modules relative to a semidualizing module and related topics. Firstly, we study [Formula: see text]-projective dimensions and [Formula: see text]-projective modules under change of rings. Secondly, we establish a new version of the Foxby equivalence with respect to [Formula: see text]-projective modules and [Formula: see text]-injective modules. Thirdly, we characterize Ding projective modules in [Formula: see text] and Ding injective modules in [Formula: see text]. At last, as applications, some new characterizations of perfect rings and quasi-Frobenius rings are given.

Cotorsion dimensions relative to semidualizing modules

Let [Formula: see text] be a semidualizing [Formula: see text]-module, where [Formula: see text] is a commutative ring. We first introduce the definition of [Formula: see text]-cotorsion modules, and obtain the properties of [Formula: see text]-cotorsion modules. As applications, we give some new characterizations for perfect rings. Second, we study the Foxby equivalences between the subclasses of the Auslander class and that of the Bass class with respect to [Formula: see text]. Finally, we discuss [Formula: see text]-cotorsion dimensions and investigate the transfer properties of strongly [Formula: see text]-cotorsion dimensions under almost excellent extensions.

Cartan-Eilenberg projective, injective and flat complexes

Let R be an associative ring with identity and F a class of R-modules. In this article: we first give a detailed treatment of Cartan-Eilenberg F complexes and extend the basic properties of the class F to the class CE(F). Secondly, we study and give some equivalent characterizations of Cartan-Eilenberg projective, injective and flat complexes which are similar to projective, injective and flat modules, respectively. As applications, we characterize some classical rings in terms of these complexes, including coherent, Noetherian, von Neumann regular rings, QF rings, semisimple rings, hereditary rings and perfect rings.

A Family of Examples of Generalized Perfect Rings

We construct a family of semiprimitive and non von Neumann regular rings satisfying that any right or left module is isomorphic to a quotient of its flat cover (in the sense of Enochs) by a small submodule. This answers in the negative a question posed by A. Amini, B. Amini, M. Ershad, and H. Sharif in 2007.

A generalization of supplemented modules

Let $M$ be a left module over a ring $R$ and $I$ an ideal of $R$. $M$ is called an $I$-supplemented module (finitely $I$-supplemented module) if for every submodule (finitely generated submodule) $X$ of $M$, there is a submodule $Y$ of $M$ such that $X+Y=M$, $X\cap Y\subseteq IY$ and $X\cap Y$ is PSD in $Y$. This definition generalizes supplemented modules and $\delta$-supplemented modules. We characterize $I$-semiregular, $I$-semiperfect and $I$-perfect rings which are defined by Yousif and Zhou [15] using $I$-supplemented modules. Some well known results are obtained as corollaries.

不均一質量とフードダンパによるインバータモータの制振

This paper deals with the quenching problem of electromagnetic vibration of the inverter motor stator. The iron stator of the inverter motor is modeled by a perfect circular ring. Single or two Houde dampers are installed on the outside of the circular ring to quench the forced vibration caused by the rotating distributed electromagnetic force. Moreover, single mass is set also on the circular ring. The solutions of forced vibration are obtained by the theoretical analysis. The following were made clear; (1) When the angle between two Houde dampers is equal to that between loop and node of the mode to be quenched, the resonant amplitude of the stator is reduced most. (2) By setting not only two Houde dampers in the suitable positions, the setting angles are apart almost 35°～65°each other, but also setting single mass which is imperfection for perfect ring, the vibration of motor stator is quenched over wide frequency region around resonance. (3) By installing single Houde damper and single mass, the vibration of motor stator is quenched over wide frequency region around resonance also. (4) The results by the ring theory coincide with those by finite element method.

Multi-scale ordered patterns in photosensitive ternary polymer mixtures

Multi-scale ordered patterns of multi-component polymer mixtures can reveal many peculiar chemical and physical properties, which makes these systems have very important potential applications in materials engineering. Via computer simulation, we create interesting ordered multi-scale structures in photosensitive and immiscible polymer mixtures. The system that we employed comprises a ternary, molten A/B/C polymer blends and the three components are mutually immiscible. Polymer C is non-optically active, while polymers A and B can exhibit reversible chemical reaction A ⇆ B induced by light. Firstly, we investigate the phase behavior of the ternary blend guided by cross-stripy mask and light, and find that a chessboard-like ordered pattern forms in the mixture before removing the mask. In the illuminated regions, the A and C components gather into ellipsoidal core-shell structures in the uncrossed illuminated area, while the A and B components gather into star structures in the crossed stripes regions. When we remove the mask, the entire system becomes illuminated, and the reaction A ⇆ B occurs throughout the film: the ellipsoidal core-shell structures of A and C components turn to spherical structures, and the star structures of A and B components turn into concentric square ring structures. Then we show the influences of the number of cross stripes and the initial composition on the formation of structure. The average spatial volume fraction of C component first increases and then decreases with the stripy number increasing and the C component net lattices play an important role in the stability of ordered structures. Secondly, when the blend is covered by the annular mask, the C component gathers to the illuminated regions and the A and B components are in radial arrangement. As a result, the mixture forms an interesting dartboard-like pattern. However, when the mask is removed, the photochemical reactions occur in the A and B components of the whole region, the increasing of free energy induces the dartboard-like pattern to be broken and to change into dots-ring structure and then it forms a perfect concentric ring pattern and the target-like pattern. And also, we show the effects of initial composition ratio of C component, the distance between two adjacent rings D, the ring width d, and the illumination intensity on the evolution of ordered structure. The larger the initial composition ratio of C component, the more easily the ordered target-like pattern forms; the larger the distance D and the smaller the width d, the better the pinning effect of C component is. The illumination intensity has little influence on the ordered morphology of the ternary system. We provide a simple approach to creating multi-scale patterned films with long-range order, which could guide us in fabricating nanoscale devices.

QUANTIFICATION OF INSTALLATION INACCURACIES AND THEIR EFFECT ON THE DESIGN OF SEGMENTAL TUNNEL LININGS

The design of segmental tunnel linings requires a clear identification of loads during construction. Displacement monitoring of the Liefkenshoek tunnel in Belgium has shown important deformations occur in the first week after installation. Afterwards no significant deformations are measured. The manually-operated segment installation causes significant imperfections in joints between segments. These remain present during the lifetime of the structure, permanently disrupting the natural force transfer between individual tunnel segments. The joint irregularities obtained from ovalization measurements were used as input in an FE-model. Calculations showed that the resulting deformations do not correspond to the classical predictions. The installation imperfections cause a transfer of normal forces to the neighboring rings via friction in the ring joints. At the same time the bending moments show a distinctive disturbance in respect to the perfect tunnel ring. These results prove that the effect of installation imperfections should not be neglected in the design of segmental tunnel linings.

A conception of zero-divisor graph for categories of modules

We introduce and study zero-divisor graphs in categories of left modules over a ring R, i.e. R- MOD . The vertices of Γ(R- MOD ) consist of all nonzero morphisms in R- MOD which are not isomorphisms. Two vertices f and g are adjacent if f ◦ g = 0 or g ◦ f = 0. We observe that these graphs are connected and their diameter is equal or less than four. We prove that Γ(R- MOD ) = 3 if and only if R is a right and left perfect ring and R/J(R) is simple artinian. We also characterize all vertices with complements and that when a kernel or a co-kernel can be a complement for a morphism. Some discussions will be made on radius of these graphs, their clique and chromatic numbers.

RETRACTION NOTE TO: X-LIFTING MODULES OVER RIGHT PERFECT RINGS

Keskin and Harmanci defined the family B(M, X) = {A ≤ M | ∃Y ≤ X, ∃f ∈ HomR(M, X/Y ), Ker f/A ? M/A}. And Orhan and Keskin generalized projective modules via the class B(M, X). In this note we introduce X-local summands and X-hollow modules via the class B(M, X). Let R be a right perfect ring and let M be an X-lifting module. We prove that if every co-closed submodule of any projective module P contains Rad(P ), then M has an indecomposable decomposition. This result is a generalization of Kuratomi and Chang’s result [9, Theorem 3.4]. Let X be an R-module. We also prove that for an X-hollow module H such that every non-zero direct summand K of H with K ∈ B(H, X), if H ⊕H has the internal exchange property, then H has a local endomorphism ring.

On the Witt Vectors of Perfect Rings in Positive Characteristic

The purpose of this article is to prove some results on the Witt vectors of perfect F p -algebras. Let A be a perfect F p -algebra for a prime integer p, and assume that A has the property P. Then does the ring of Witt vectors of A also have P? A main theorem gives an affirmative answer for P = ″integrally closed” under a very mild condition.

Direct sums of ADS* modules

A module M is called ADS* if for every direct summand N of M and every supplement K of N in M, we have \(M=N\oplus K\). In this work, we study direct sums of ADS* modules. Many examples are provided to show that this notion is not inherited by direct sums. It is shown that if a module M has a decomposition \(M=A\oplus B\) which complements direct summands such that A and B are mutually projective, then M is ADS*. The class of rings R, for which all direct sums of ADS* R-modules are ADS*, is shown to be exactly that of the right V-rings. We characterize the class of right perfect rings R for which \(R\oplus S\) is ADS* for every simple R-module S as that of the semisimple rings.

Transmission of light through double gold nanobars embedded in split ring pair array

The optical transmission of a normal incident wave through double nanobars embedded in split ring pair (DNSRP) array was investigated by using the finite difference time domain (FDTD) method. It is demonstrated that, compared to double nanobars embedded in perfect ring pair structure, the transmission spectra of DNSRP array showed more dips and peaks. Along with the changes of slit widths, distances between the double nanobars and the environmental dielectric constant, the optical and plasmonic characteristics of the DNSRP array were tuned. Based on the electric field distributions, we discuss the physical mechanism of light passing through the DNSRP array.

Photodetachment of negative hydrogen ion in circularly polarized monochromatic and bichromatic laser fields

We theoretically have investigated the detachment of negative hydrogen ion in circularly polarized monochromatic and bichromatic laser fields by calculating the photoelectron momentum distributions (PMDs). We find that the PMDs are controlled by the electric field of laser pulse. For circularly polarized monochromatic laser fields, the PMDs are perfect and isotropic rings, and the photodetachment rates in random direction are same. For this case, the photoelectron momentum spectra directly reflect the electronic distributions of negative hydrogen ion. For circularly polarized bichromatic laser field, the PMDs are inversion asymmetric, but symmetric about a certain axis. The emission directions of the photoelectrons vary with the electric field shape, that is to say that the emission directions of the photoelectrons vary with the carrier-envelope (CE) phases, which decide the electric field of laser pulse. The maximal photodetachment rates and the spectra shape, however, are same for the different CE phases.

Σ‐algebraically compact modules and ‐compact cardinals

We prove that the property characterizes Σ‐algebraically compact modules if is not ω‐measurable. Moreover, under a large cardinal assumption, we show that over any ring R where is not ω‐measurable, any free module M of ω‐measurable rank satisfies , hence the assumption on cannot be dropped in general (e.g., over small non‐right perfect rings). In this way, we extend results from a recent paper by Simion Breaz .