Perfect Rings Research Articles

Effect on Au adatom of the different configurations of BnNn (n = 1–3) co-doped graphene

The effects of different amounts of BN atom pairs on the structural and electronic properties of doped graphene, as well as the adsorption of Au atoms, were studied by first-principles calculations. The results show that the first-nearest-neighbor structure of B and N atoms was the most stable, and perfect BN hexagon rings can be substituted for BNC ones with increasing number of BN atom pairs in BN co-doped graphene. In the co-doped graphene, the optimal adsorption site for Au atoms was on top of a B atom. The site on the B atom between N and C atoms was more stable than that between N and N atoms. All optimal configurations of BnNn co-doped graphene were semiconductors, which turned into metals when an Au atom was adsorbed. These results are of great significance in guiding new research on the adsorption of BN co-doped graphene with the selection of doping configurations.

Model structures, n-Gorenstein flat modules and PGF dimensions

AbstractGiven a non-negative integer n and a ring R with identity, we construct a hereditary abelian model structure on the category of left R-modules where the class of cofibrant objects coincides with $\mathcal{GF}_n(R)$ the class of left R-modules with Gorenstein flat dimension at most n, the class of fibrant objects coincides with $\mathcal{F}_n(R)^\perp$ the right ${\rm Ext}$-orthogonal class of left R-modules with flat dimension at most n, and the class of trivial objects coincides with $\mathcal{PGF}(R)^\perp$ the right ${\rm Ext}$-orthogonal class of PGF left R-modules recently introduced by Šaroch and . The homotopy category of this model structure is triangulated equivalent to the stable category $\underline{\mathcal{GF}(R)\cap\mathcal{C}(R)}$ modulo flat-cotorsion modules and it is compactly generated when R has finite global Gorenstein projective dimension.The second part of this paper deals with the PGF dimension of modules and rings. Our results suggest that this dimension could serve as an alternative definition of the Gorenstein projective dimension. We show, among other things, that (n-)perfect rings can be characterized in terms of Gorenstein homological dimensions, similar to the classical ones, and the global Gorenstein projective dimension coincides with the global PGF dimension.

Relative injective modules, superstability and noetherian categories

We study classes of modules closed under direct sums, [Formula: see text]-submodules and [Formula: see text]-epimorphic images where [Formula: see text] is either the class of embeddings, RD-embeddings or pure embeddings. We show that the [Formula: see text]-injective modules of theses classes satisfy a Baer-like criterion. In particular, injective modules, RD-injective modules, pure injective modules, flat cotorsion modules and [Formula: see text]-torsion pure injective modules satisfy this criterion. The argument presented is a model theoretic one. We use in an essential way stable independence relations which generalize Shelah’s non-forking to abstract elementary classes. We show that the classical model theoretic notion of superstability is equivalent to the algebraic notion of a noetherian category for these classes. We use this equivalence to characterize noetherian rings, pure semisimple rings, perfect rings and finite products of finite rings and artinian valuation rings via superstability.

On $G$-$(n,d)$-rings and $n$-coherent rings

Let $n$ and $d$ be non-negative integers. We introduce the concept of $strongly$ $(n,d)$-$injective$ modules to characterize $n$-coherent rings. For a right perfect ring $R$, it is shown that $R$ is right $n$-coherent if and only if every right $R$-module has a strongly $(n,d)$-injective (pre)cover for some non-negative integer $d \leq n$. We also provide equivalent conditions for an $(n,d)$-ring being $n$-coherent. Then we investigate the so-called $right$ $G$-$(n,d)$-$rings$, over which every $n$-presented right module has Gorenstein projective dimension at most $d$. Finally, we prove a Gorenstein analogue of Costa's first conjecture.

When every S-flat module is (flat) projective

Let R be a commutative ring with identity and S a multiplicative subset of R. The aim of this paper is to study the class of commutative rings in which every S-flat module is flat (resp., projective). An R-module M is said to be S-flat if the localization of M at S, MS , is a flat RS -module. Commutative rings R for which all S-flat R-modules are flat are characterized by the fact that R/Rs is a von Neumann regular ring for every s ∈ S . While, commutative rings R for which all S-flat R-modules are projective are characterized by the following two conditions: R is perfect and the Jacobson radical J(R) of R is S-divisible. Rings satisfying these conditions are called S-perfect. Moreover, we give some examples to distinguish perfect rings, S-perfect rings, and semisimple rings. We also investigate the transfer results of the “S-perfectness” for various ring constructions, which allows the construction of more interesting examples.

Effects of Initial Small-Scale Material Nonlinearity on the Pre-Yield and Pre-Buckling Response of an Externally Pressurized Ring

The effects of initial small-scale material nonlinearity on the pre-yield and pre-buckling response of externally pressurized metallic (plane strain) perfect rings (very long cylindrical shells) is investigated. The cylindrically curved 16-node element, based on an assumed quadratic displacement field (in surface-parallel coordinates) and the assumption of linear distribution of displacements through thickness (LDT), is employed to obtain the discretized system equations. The effect of initial small-scale material nonlinearity (assumed hypo-elastic) on the deformation and stress in the pre-yield and pre-buckling regime of a very long relatively thin metallic cylindrical shell (plane strain ring) is numerically investigated. These numerical results demonstrate that the enhanced responses for metallic rings due to initial small-scale nonlinearity are significant enough to not miss attentions from designers and operators of submersibles alike.

Rings Whose Certain Modules Are Dual Self-CS-Baer

In this work, we characterize some rings in terms of dual self-CS-Baer modules (briefly, dsCS-Baer modules). We prove that any ring $\Sigma$ is a left and right artinian serial (briefly, as-) ring with $J^2=0$ iff $\Sigma\oplus Z$ is dsCS-Baer for every right $\Sigma$-module $Z$. If $\Sigma$ is a commutative ring, then we prove that $\Sigma$ is an as-ring iff $\Sigma$ is perfect and every $\Sigma$-module is a direct sum of (cyclic) dsCS-Baer $\Sigma$-modules. Also, we show that $\Sigma$ is a right perfect ring iff all countably generated free right $\Sigma$-modules are dsCS-Baer.

High adherence to intermittent and continuous use of a contraceptive vaginal ring among women in a randomized controlled trial in Kigali, Rwanda.

The introduction of female-initiated drug-delivery methods, including vaginal rings, have proven to be a promising avenue to address sexually transmitted infections and unintended pregnancies, which disproportionally affects women and girls in sub-Saharan Africa. Efficient uptake of existing and new technologies such as vaginal rings requires in depth understanding of product adherence. This remains a major challenge as data on adherence to vaginal rings from African countries is limited. In this study, we explored adherence of contraceptive vaginal ring (NuvaRing®) use in Kigali, Rwanda using a mixed methods approach. We collected quantitative and qualitative data at multiple time points from women participating in a clinical trial exploring the safety and acceptability of either intermittent or continuous use of the NuvaRing®. Various adherence categories were used including monthly and cumulative adherence measurement. The quantitative data were analysed using R and the qualitative data were analysed using a deductive, content-analytical approach based on categories related to the quantitative adherence measures. All data were compared and triangulated. Data from 120 enrolled participants showed that self-reported adherence was high at every study visit in both study groups. At first study visit 80% of the intermittent ring users and 79.7% of the continuous ring users reported perfect adherence (assessed as "the ring was never out"). Reporting of ring expulsions and removals were highest (28.3%) at the beginning of the trial. Self-reported perfect ring adherence increased during the study and reports of ring expulsions and removals declined as familiarity with this contraceptive method increased. The percentage of women with perfect cumulative adherence was non-significantly higher in the intermittent (61.7%) than in the continuous use group (54.3%). The low rate of discrepant adherence data after triangulation (6%) is in line with the perception of the participants as adherent throughout the study. Self-reported adherence in both study groups was high with removals and expulsions being within the expected product range. Comprehensive adherence data triangulation allowed for a deeper understanding of context-driven behaviour that shaped adherence patterns and challenges. Our data categorisation and triangulation approach has shown potential for implementation in future vaginal ring studies aiming to better understand and measure adherence.

Non-Templated Assembly of D5h-Symmetric Pd5L10 Rings by Precise Ligand Angle Adjustment.

We report a series of Pd(II)nL2n coordination rings for which nuclearity is controlled by the binding angle of the corresponding bis-monodentate bridging ligands. Judicious choice of the angle within a family of rather rigid ligands allowed for the first-time to synthesize a homoleptic five-membered Pd5L10 ring that does not require any template to form. We demonstrate that control over the ring size is maintained both in the solid-, solution-, and gas-phase. Two X-ray structures of five-membered rings from ligands with ideal angles (yielding a perfect pentagonal ring) vs. suboptimal angles (resulting in a highly distorted structure) illustrate the importance of the correct ligand geometry. A mathematical model for estimating the expected ring size based on the ligand angle was derived and DFT computations show that ring-strain is the major factor determining the assembly outcome.

On Perfect Rings Dominations in Graphs

On Perfect Rings Dominations in Graphs

Concerning the Nakayama property of a module

Abstract In this paper, we thoroughly study the Nakayama property and some related concepts. Also, we describe multiplication modules that, among other things, satisfy the Nakayama property. Next, we show that a ring 𝑅 is a Max ring if and only if all modules that can be generated by a finite or countable set have the weak Nakayama property. We prove that a ring 𝑅 is a perfect ring if and only if every module that can be generated by a finite or countable set has the Nakayama property. Finally, we present some categorical results on the aforementioned properties.

Sincere silting modules and vanishing conditions

We study characterizations of sincere modules, sincere silting modules and tilting modules in terms of various vanishing conditions. Let R be a perfect ring and T be an R-module. It is proved that T is sincere silting if and only if T is presilting satisfing the vanishing condition KerExt R 0 ≤ i ≤ 1 ( T , − ) = 0 , and that T is tilting if and only if KerExt R 0 ⩽ i ⩽ 1 ( T , − ) = 0 and Gen T ⊆ KerExt R 1 ⩽ i ⩽ 2 ( T , − ) . As an application, we prove that a sincere silting R-module T of finite projective dimension is tilting if and only if Ext R i ( T , T ( J ) ) = 0 for all sets J and all integer i ≥ 1 . This not only extends a main result of Zhang’s paper [Self-orthogonal τ-tilting modules and tilting modules, J Pure Appl Algebra, 2022, 226: 106860] from finitely generated modules over Artin algebras to infinitely generated modules over more general rings, but also gives it a different proof without using Auslander-Reiten translations.

Rings close to periodic with applications to matrix, endomorphism and group rings

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl’s question whether or not a ring R being nil-clean implies that the matrix ring M n ( R ) over R is also nil-clean for all n ≥ 1 is paralleling to the corresponding implication for (abelian, local) periodic rings. Besides, we study when the endomorphism ring E ( G ) of an abelian group G is periodic. Concretely, we establish that E ( G ) is periodic exactly when G is finite as well as we find a complete necessary and sufficient condition when the endomorphism ring over an abelian group is strongly m-nil clean for some natural number m thus refining an “old” result concerning strongly nil-clean endomorphism rings. Responding to a question when a group ring is periodic, we show that if R is a right (resp., left) perfect periodic ring and G is a locally finite group, then the group ring RG is periodic, too. We finally find some criteria under certain conditions when the tensor product of two periodic algebras over a commutative ring is again periodic. In addition, some other sorts of rings very close to periodic rings, namely the so-called weakly periodic rings, are also investigated.

Compairing Categories of Lubin–Tate (varphi _L,Gamma _L)-Modules

In the Lubin–Tate setting we compare different categories of (varphi _L,Gamma )-modules over various perfect or imperfect coefficient rings. Moreover, we study their associated Herr-complexes. Finally, we show that a Lubin Tate extension gives rise to a weakly decompleting, but not decompleting tower in the sense of Kedlaya and Liu.

On the strict Picard spectrum of commutative ring spectra

We compute the connective spectra of maps from $\mathbb {Z}$ to the Picard spectra of the spherical Witt vectors associated with perfect rings of characteristic $p$. As an application, we determine the connective spectrum of maps from $\mathbb {Z}$ to the Picard spectrum of the sphere spectrum.

Commutative ring extensions defined by perfect-like conditions

UDC 512.5 In 2005, Enochs, Jenda, and López-Romos extended the notion of perfect rings to n -perfect rings such that a ring is n -perfect if every flat module has projective dimension less or equal than n . Later, Jhilal and Mahdou defined a commutative unital ring R to be strongly n -perfect if any R -module of flat dimension less or equal than n has a projective dimension less or equal than n . Recently Purkait defined a ring R to be n -semiperfect if R ¯ = R / R a d ( R ) is semisimple and n -potents lift modulo R a d ( R ) . We study of three classes of rings, namely, n -perfect, strongly n -perfect, and n -semiperfect rings. We investigate these notions in several ring-theoretic structures with an aim of construction of new original families of examples satisfying the indicated properties and subject to various ring-theoretic properties.

Coretractable modules relative to a module

In this paper, we introduce and study a generalization of coretractable modules. An [Formula: see text]-module [Formula: see text] is called coretractable relative to an R-module N if [Formula: see text], for any proper submodule [Formula: see text] of [Formula: see text]. It is shown that a right [Formula: see text]-module [Formula: see text] is coretractable relative to [Formula: see text] if and only if [Formula: see text] is coretractable relative to [Formula: see text], for some [Formula: see text]. Also we show that if [Formula: see text] is a right quasi-duo ring, then there exists a right [Formula: see text]-module that it is coretractable relative to [Formula: see text], for any nonzero right [Formula: see text]-module [Formula: see text] if and only if [Formula: see text] is a left perfect local ring.

Research on phase transition induced plastic deformation in nanoindentation of single crystal diamond

Nanoindentation experiments were performed on the {100} crystal plane of single crystal diamond using molecular dynamics simulation to study the mechanism of its plastic deformation. The relationship between phase transition and plastic deformation during the nanoindentation process was discussed using the load-displacement (P–h) curve and phase transition evolution process obtained from the experiment. It was observed that the initial plastic characteristic of diamond was because of the phase transition, rather than dislocation slip. The nucleation and growth of 1/2[110]{111} perfect dislocation ring during the plastic deformation of single crystal diamond were determined by observing pop-in events and dislocation evolution in P-h curves. The plastic deformation of single crystal diamond was attributed to the formation and growth of 1/2[110]{111} complete dislocation ring caused by the phase transition from disordered sp3 to ordered sp2 carbon atoms based on the further analysis of microscopic defects. It was considered that the phase transition during the nanoindentation of single crystal diamond is an important excitation source for plastic deformation at the atomic scale. Finally, it was observed through the analysis of bond length and angle during the phase transition process that the CC bond fracture of diamond was because of the transformation from sp3 to sp2 carbon atoms, and the fracture and recombination of adjacent carbon atoms, which led to graphitization. This conclusion confirmed that graphitization is the mechanism of plastic deformation of diamond and the pop-in event is caused by the stress release caused by the rupture of CC bond.

RD-flatness and RD-injectivity of simple modules

We say that a ring R is a right RDV-ring if each simple right R -module is RD -injective. In this paper, we study the notion of RDV -rings which is a non-trivial generalization of V-rings and Köthe rings. For instance, commutative RD -rings, serial rings and right duo right uniserial rings are RDV -rings. Several characterizations of right RDV -rings are given. Also, it is shown that over a semilocal ring R with Jacobson radical J , each simple right R -module is RD -flat if and only if R is a left RDV -ring, if and only if R ( R / J ) is RD -injective, if and only if ( R / J ) R is RD -flat. As a consequence, we show that a local ring R is a principal ideal ring if and only if R satisfies the ascending chain condition on principal left ideals and R ( R / J ) is RD -injective. In the case of R being either a local left perfect ring or a normal left perfect ring, we have obtained results which state that to check whether every left R -module is RD -injective (or, R is Köthe), it suffices to test only the RD -injectivity of the simple left R -modules. Finally, we give some characterizations of quasi-Frobenius rings by using these concepts.

Characterization of rings whose classical quotient rings are perfect rings

Characterization of rings whose classical quotient rings are perfect rings