Ring Theory Research Articles

A Survey on Ring Theory for its Mathematical Foundation and Applications

The origin of the ring theory can be dated to the early 19th century. With the rapid development of technology, more and more complex engineering problems and computer problems need to obtain support and results from more basic mathematics theory. Hence, based on this demand, the researcher discovered that ring theory is a very noteworthy math theory for solving the complicated problems above. This paper investigated the ring theory and related articles based on number theory and corresponding applications, which produces and results in a literature review in this particular direction.

Bipolar complex fuzzy near rings

In this article, we devise the novel concept of bipolar complex fuzzy (BCF) near rings (BCFNR), to fill a momentous research gap in connection of bipolar complex fuzzy sets with the theory of near rings in the existing literature. We expand the theory of near ring into the structure of BCF set (BCFS), offering a more suitable approach for the representation of algebraic systems with inherent ambiguity, bipolarity, and 2nd dimension information. Further, we introduce the concept of the bipolar complex fuzzy sub-near ring (BCFSNR), bipolar complex fuzzy left ideal (BCFLI), bipolar complex fuzzy right ideal (BCFRI), and bipolar complex fuzzy ideal (BCFI) in the near ring. After that, we prove the related theorem and results of these devised concepts. We also introduce theorems based on the homomorphism theory and Noetherian theory of near rings within the frame of bipolar complex fuzzy near rings. At the end of the manuscript, we reveal the application of BCFNRs in decision-making (DM) and illustrate it through an example.

Molecular Topology Index of a Zero Divisor Graph on a Ring of Integers Modulo Prime Power Order

In chemistry, graph theory has been widely utilized to address molecular problems, with numerous applications in graph theory and ring theory within this field. One of these applications involves topological indices that represent chemical structures with numerical values. Various types of topological indices exist, including the Wiener index, the first Zagreb index, and the hyper-Wiener index. In the context of this research, the values of the Wiener index, the first Zagreb index, and the hyper-Wiener index for zero-divisor graphs on the ring of integers modulo a prime power order will be explored through a literature review and conjecture.

Analysis and evaluation of multi-state wear mechanism of elastic-flexible thin-walled bearings

EFTWBs (Elastic-flexible thin-walled bearings) are mainly used in harmonic reducers. Under the condition of large reduction ratio and cyclic rotation, LFW (linear fretting wear), CFW (curve fretting wear) and CRF (complete rolling friction) appear in the bearing raceway. In this study, LFW and CFW experiments were carried out, and the relationship between coefficient of friction and surface hardness was analyzed. Meanwhile, the multi-dimensional analysis (surface hardness, residual stress, surface/subsurface retained austenite) of the raceway surface integrity of EFTWBs after 8000 h of high service was analyzed. Combined with the manufacturing process of EFTWBs and the above conclusions, the multi-state wear mechanism of raceway was comprehensively evaluated. The evaluation conclusions show that the raceway of EFTWBs contained CRF (including pitting and spalling) and sliding fretting wear (including LFW and CFW). The axial and tangential residual stresses of the untested and tested outer and inner ring raceways were compressive stresses. Grain delamination appeared on the tested outer and inner ring raceway subsurface (20 µm from the surface). The wear of the outer ring raceway of EFTWBs was greater than that of the inner ring, and the wear was multiple. A 430 nm nanocrystalline layer was found under FIB-TEM on the surface layer of the outer ring raceway, and the fracture and obvious dislocation appeared. The rolling-sliding wear mechanism of EFTWBs raceway was comprehensively evaluated. The accuracy of this analysis was verified by combining thin-walled ring and Hertz theories. There are many detailed conclusions which provide theoretical support and data analysis for relevant manufacturers of harmonic reducer and robot EFTWBs.

Caring for terminally Ill patients: the impact on oncologists

BackgroundJourneying with patients throughout their cancer trajectory and caring for them at the end of life can lead to emotional and moral distress in oncologists, negatively impacting their personal and professional identities. A better understanding of how transitions in care goals affect oncologists can shed light on the challenges faced and the support required. This study explored the impact of care transitions on oncologists’ professional identity formation (PIF).MethodsFrom September to December 2023, semi-structured interviews were conducted with oncologists in a palliative care center in Singapore. The Ring Theory of Personhood (RToP) was used as a framework to capture the effects of experiences with patients transitioning from curative to palliative care on the oncologists’ sense of self and identity. Data were analyzed using both inductive and deductive qualitative analysis.ResultsParticipants included six female and six male physicians, aged 30 to 53 years (mean 38 years), with an average of 9.75 years of experience as oncologists. The main domains identified were 1) challenges faced in transitioning patients to palliative care, 2) the impact of dealing with dying patients on oncologists, and 3) coping mechanisms.ConclusionOncologists experience self-doubt and moral distress as they manage transitions in care. The PIF of oncologists can be supported through reflection and introspection, peer support, and interventions to promote self-care — ultimately enabling them to make meaning of their experiences, renew family ties, and reaffirm their commitment to the profession.

On the ϕ-weak global dimensions of polynomial rings and ϕ-Prüfer rings

This paper focuses on the study of [Formula: see text]-weak global dimensions in the context of polynomial rings and [Formula: see text]-Prüfer rings. We explore new properties of these dimensions and extend the Hilbert syzygy theorem to [Formula: see text]-weak global dimensions of rings. We also determine the [Formula: see text]-weak global dimension for certain types of [Formula: see text]-Prüfer rings. Key concepts such as [Formula: see text]-flat modules, [Formula: see text]-injective modules, and [Formula: see text]-torsion modules are discussed, along with their hereditary properties in PN-rings. This paper includes several theorems and lemmas that provide insights into the [Formula: see text]-weak global dimensions and their implications in the field of ring theory.

On fusible rings and related notions

A ring is called left fusible if each of its nonzero elements is the sum of a left zero-divisor and a non-left zero-divisor. This paper aims to extend the existing theory of left fusible rings and related notions such as being left unit-fusible, being regular left fusible, and being uniquely left fusible. New examples of these rings are presented. A new notion of left generalized unit-fusible rings is introduced and discussed. The asymmetry of these notions is addressed.

Max(dL) revisited

This article studies different topological properties of the space of maximal d-elements of an M-frame with a unit. We characterize when the space Max(dL) is Hausdorff, answering the question posed in [2]. We also characterize other topological properties of Max(dL), namely zero-dimensional, discrete, and clopen π-base. The concept of weak-component elements is introduced here, as a generalized idea from the theory of rings, which is essential in the study of d-semiprime frames.

Some results on the total zero-divisor graph of a commutative ring

PurposeThe purpose of this paper is to characterize a commutative ring R with identity which is not an integral domain such that ZT(R), the total zero-divisor graph of R is connected and to determine the diameter and radius of ZT(R) whenever ZT(R) is connected. Also, the purpose is to generalize some of the known results proved by Duric et al. on the total zero-divisor graph of R.Design/methodology/approachWe use the methods from commutative ring theory on primary decomposition and strong primary decomposition of ideals in commutative rings. The structure of ideals, the primary ideals, the prime ideals, the set of zero-divisors of the finite direct product of commutative rings is used in this paper. The notion of maximal Nagata prime of the zero-ideal of a commutative ring is also used in our discussion.FindingsFor a commutative ring R with identity, ZT(R) is the intersection of the zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R. The zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R are well studied. Hence, we determine necessary and sufficient condition so that ZT(R) agrees with the zero-divisor graph of R (respectively, agrees with the total graph induced by the set of non-zero zero-divisors of R). If Z(R) is an ideal of R, then it is noted that ZT(R) agrees with the zero-divisor graph of R. Hence, we focus on rings R such that Z(R) is not an ideal of R. We are able to characterize R such that ZT(R) is connected under the assumptions that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. With the above assumptions, we are able to determine the domination number of ZT(R).Research limitations/implicationsDuric et al. characterized Artinian rings R such that ZT(R) is connected. In this paper, we extend their result to rings R such that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. As an Artinian ring is isomorphic to the direct product of a finite number of Artinian local rings, we try to characterize R such that ZT(R) is connected under the assumption that R is ta finite direct product of rings R1, R2, … Rn with Z(Ri) is an ideal of Ri for each i between 1 to n. Their result on domination number of ZT(R) is also generalized in this paper. We provide several examples to illustrate our results proved.Practical implicationsThe implication of this paper is that the existing result of Duric et al. is applicable to large class of commutative rings thereby yielding more examples. Moreover, the results proved in this paper make us to understand the structure of commutative rings better. It also helps us to learn the interplay between the ring-theoretic properties and the graph-theoretic properties of the graph associated with it.Originality/valueThe results proved in this paper are original and they provide more insight into the structure of total zero-divisor graph of a commutative ring. This paper provides several examples. Not much work done in the area of total zero-divisor graph of a commutative ring. This paper is a contribution to the area of graphs and rings and may inspire other researchers to study the total zero-divisor graph in further detail.

DERIVATION ON SEVERAL RINGS

Research on ring derivation is one of the studies that is quite popular among algebra lovers. The definition of the derivation on the ring is motivated by the derivation in calculus which has Leibniz's rule. The purpose of this paper is to show some of the derivation properties on several rings, namely divisor rings, cartesian product rings, and factor rings. Let be a commutative ring with multiplicative identity and A the set of multiplicative closed that has non-zero divisor. In this paper, we have shown some results of derivation on ring theory. If is a ring derivation of R and is a divisor ring of , we can construct for all , then the map is a derivation on . The concept of embedding one ring into another ring can be used so that the ring of constant of , namely , is a subring of the divisor ring . Related to the ideal on ring theory, if I is an ideal of R, then where is also a derivation on the ring . The last result in this paper comes from the ring of cartesian product, take be a ring with derivation for . The cartesian product ring have a derivation ring defined by for any .

Structure of zero-divisors to go up to related ideals

There are many ways for zero-dividing polynomials to go up to zero-dividing ideals when the base rings are IFP. The importance of these in ring theory leads us to consider the following ring conditions and study new useful roles of matrices for ring theory. Let R be a ring and a , b ∈ R \\ { 0 } . The first is the condition (*) that if ab = 0 then Ib = 0 for some nonzero ideal I ⊆ RaR of R or aJ = 0 for some nonzero ideal J ⊆ RbR of R. It is shown that from given any IFP ring, there can be constructed a non-IFP ring with the condition (*). We prove that a semiprime ring R with the condition (*) is both right and left nonsingular. The second is the condition (**) that if ab = 0 then IJ = 0 for some nonzero ideals I ⊆ RaR and J ⊆ RbR of R. We prove that every ring can be a subring of rings with the condition (**), that if R is an irreducible ring with the condition (**) then R is either a domain or non-semiprime, and that the condition (**) passes to polynomial rings when the base ring is semiprime. Various sorts of examples are given to illustrate and delimit the results obtained.

Latent profiles of narrative competence and professional identity among nursing students: A cross-sectional analytic study based on the Ring theory of personhood

AimTo identify latent profiles of narrative competence in nursing students and examine the association between the potential competence profiles and professional identity from a person-centred perspective. BackgroundAccording to the Ring theory of personhood, nursing students can develop their professional identities from individual, relational and social aspects through interaction with patients, as well as listening to, understanding and responding to patients' disease narratives. However, few studies have examined the relationship between narrative competence and professional identity through the quantitative method. DesignA cross-sectional analytic study. MethodsA total of 472 nursing students responded to the survey between March and May 2023. The Professional Identity Questionnaire for Nurse Students and the Narrative Competence Scale were given to participants. Latent profile analysis was conducted to identify narrative competence profiles. The Bolck-Croon-Hagenaars method was used to analyse whether these latent profiles for narrative competence affected nursing students’ general, individual, interpersonal and social professional identities. ResultsLatent profiles were identified as “low narrative competence” (12.1 %), “relatively low narrative competence” (39.9 %), “moderate narrative competence” (40.1 %) and “high narrative competence” (7.9 %). The profiles only show level differences rather than combinations of competence areas. These profiles had varying effects on the nursing students’ general professional identities, as well as their individual, relational and social professional identities. ConclusionThis study highlights the significance of providing tailored guidance and support to nursing students, taking into account their unique narrative competency profile, to promote the formation of professional identity from individual, relational and social aspects. Nursing educators should effectively distinguish nursing students with inadequate narrative competence and value patients’ disease narratives to promote narrative competence and professional identity.

On center graphs of finite associative rings

UDC 512.5 We consider a finite associative ring R , which may have or may not have a unit element. We also examine its center denoted by Z ( R ) . Our main focus is on the introduction of two distinct graphs associated with R , namely, the center graph denoted by G C ( R ) and the strict center graph denoted by G C ( R ) ¯ . We present the properties of G C ( R ) and explore its implications on the nature of Z ( R ) . Specifically, we demonstrate that if G C ( R ) is complete, then Z ( R ) is an ideal in R . Furthermore, in the case where R is a unital ring, the completeness of G C ( R ) leads to the conclusion that R is a commutative ring. As a specific application of our results, we provide an explicit construction of the graph G C ¯ ( T 2 ( p ) ) , where T 2 ( p ) represents the ring of upper-triangular matrices with entries in the ring ℤ / p ℤ and p is a prime integer. In our investigations of the center graph and strict center graph, we aim to shed light on the properties of finite associative rings and their centers, thus providing valuable insights and applications in the ring theory.

Genoarchitectural Definition of the Adult Mouse Mesocortical Ring: A Contribution to Cortical Ring Theory.

Data mining was performed at the databases of the Allen Institute for Brain Science (RRID:SCR_017001) searching for genes expressed selectively throughout the adult mouse mesocortex (transitional cortex ring predicted within the concentric ring theory of mammalian cortical structure, in contrast with central isocortex [ICx] and peripheral allocortex). We aimed to explore a shared molecular profile selective of all or most mesocortex areas. This approach checks and corroborates the precision of other previous definitory criteria, such as poor myelination and high kainate receptor level. Another aim was to examine which cortical areas properly belong to mesocortex. A total of 34 positive adult selective marker genes of mesocortex were identified, jointly with 12 negative selective markers, making a total of 46 markers. All of them identify the same set of cortical areas surrounding the molecularly different ICx as well as excluding adjacent allocortex. Four representative mesocortex markers-Crym, Lypd1, Cdh13, and Smoc2-are amply illustrated, jointly with complementary material including myelin basic protein, to check myelination, and Rorb, to check layer 4 presence. The retrosplenial (ReSp) area, long held to be mesocortical, does not share any of the 46 markers of mesocortex and instead expresses Nr4a2 and Tshz2, selective parahippocampal allocortex markers. Moreover, it is not hypomyelinic and lacks a Rorb-positive layer 4, aspects generally present in mesocortex. Exclusion of the ReSp area from the mesocortex ring reveals the latter to be closed at this locus instead by two adjacent areas previously thought to be associative visual ICx (reidentified here molecularly as postsplenial and parasplenial mesocortex areas). The concepts of ICx, mesocortex, and parahippocampal allocortex are thus subtly modified by substantial molecular evidence.

Hybrid structure of maximal ideals in near rings

A hybrid structure is an arrangement that makes use of many hierarchical reporting structures and is applied to algebraic structures such as groups and rings. In the discipline of abstract algebra, an ideal of a near-ring is a unique subset of its elements in ring theory. Ideals generalize specific subsets of integers, such as even numbers or multiples of three. Researchers have been using mathematical theories of fuzzy sets in ring theory to explain the uncertainties that emerge in various domains such as art and science, engineering, medical science, and in environment. By fusing soft sets and fuzzy sets, a new mathematical tool that has significant advantages in dealing with uncertain information is provided. Consequently, there is always some discrepancy between reality's haziness and its mathematical model's precision. Hence ring theory has been widely used in many researches but there is some uncertainty in converting the fuzzy sets to a hybrid structure of any algebraic structure. Many approaches were done in groups. Therefore, the Hybrid structure of fuzzy sets in near rings is introduced, in which the fuzzy ideals are converted to hybrid ideals and fuzzy maximal ideals are converted to hybrid maximal ideals. For hybridization, firstly the hybrid structure is established and then sub-near rings and near rings are also determined. Then the hybrid structure of sub-near rings and ideals is introduced. This converts the fuzzy ideals and fuzzy maximal ideals to hybrid ideals and hybrid maximal ideals. The result obtained by the proposed model efficiently solved the uncertainty problems and the effectiveness of the proposed approach shows the best class, mean, worst class, and time complexity.

A novel semi-analytical method for fillet foundation deflection calculation in presence of a tooth crack propagating into the rim of a spur gear

Existing analytical approaches for calculating fillet foundation deflections in spur gear struggle to adequately address the complexities introduced by tooth cracks, particularly as cracks propagate into the gear body. The finite element methods are accurate but computationally expensive. In response to these challenges, the paper introduces a novel semi-analytical method based on elastic circular ring theory that is capable of calculating the fillet foundation deflection for both healthy and cracked conditions. To establish the usefulness of the method, the fillet foundation deflections obtained are further used to obtain the time-varying mesh stiffness. A methodology centered on compatibility conditions is employed to achieve load distribution during double tooth engagement. The results are verified using two three-dimensional finite element models. In addition, the effect of different hub hole radii, crack lengths and orientations on the fillet foundation deflections and TVMS are investigated. The outcomes suggest that the proposed semi-analytical approach demonstrates superior accuracy in the calculation of spur gear mesh stiffness compared to alternative methods.

Nonlinear Dynamic Properties of Rigid–Elastic–Liquid Coupled Ball Bearings Considering External Load Excitation

In this study, a novel dynamic model of a ball bearing based on rigid–elastic–liquid (REL) coupling structure considering external load excitation is established, and the dynamic response of its outer ring under external load excitation is examined. First, the differential equation of the journal motion (i.e. the outer ring) is derived based on Newton’s law, and the oil film force induced by the REL coupling structure is determined by using the central finite difference method, planar bending theory of thin ring, and Simpson’s rule. Further, the fourth and fifth-order variable step length Runge–Kutta method is used to solve the kinematic differential equation. Second, the dynamic characteristics of the outer ring of the bearing under different speeds and elastic support structural parameters are examined. Finally, the feasibility of the theoretical model is verified through experiments. The results show that compared with the traditional squeeze film damper (SFD) system, the ball bearing with REL coupling structure is more stable, and the vibration damping effect is better under higher bearing speed, lower squirrel cage stiffness, and smaller elastic ring flexibility coefficient. The above findings provide useful insights into the dynamic characteristics of REL coupled ball bearings under unbalanced excitation, which can serve as a reference for the optimization design of such ball bearings.

Topological aspects of the ideal theory in rings of integer-valued polynomials

Let D D be a Dedekind domain with finite residue fields. We provide topological insights into certain classes of ideals of I n t ( D ) Int(D) lying over a given maximal ideal m \mathfrak m of D D . We completely determine invertible/divisorial ideals in terms of topological properties of subsets of the m \mathfrak m -adic completion of D D . Moreover, these results are naturally extended to overrings of I n t ( D ) Int(D) . As an application we provide explicit constructions of divisorial ideals of I n t ( D ) Int(D) which are not finitely generated.

Preface of the JAA Special Issue Recent Advances in Non-Commutative Ring Theory

Preface of the JAA Special Issue Recent Advances in Non-Commutative Ring Theory

Small and Simple Systems That Favor the Arrow of Time.

The 2nd law of thermodynamics yields an irreversible increase in entropy until thermal equilibrium is achieved. This irreversible increase is often assumed to require large and complex systems to emerge from the reversible microscopic laws of physics. We test this assumption using simulations and theory of a 1D ring of N Ising spins coupled to an explicit heat bath of N Einstein oscillators. The simplicity of this system allows the exact entropy to be calculated for the spins and the heat bath for any N, with dynamics that is readily altered from reversible to irreversible. We find thermal-equilibrium behavior in the thermodynamic limit, and in systems as small as N=2, but both results require microscopic dynamics that is intrinsically irreversible.