Ring Theory Research Articles

The challenge of structuralist thinking in abstract algebra: a transition problem

This paper deals with the issue of transitions in the teaching and learning of abstract algebra throughout the Bachelor in France, focusing on the notion of ideal in ring theory which is seen as a gateway to structuralist thinking. In the framework of the Anthropological Theory of the Didactic, the notion of structuralist praxeologies, endowed with structuralist levels, is used to analyse a corpus of course documents from three upper-secondary teachers which cover the teaching of ideals through the Bachelor. Hausberger's three-phase model for structuralist transitions, which generalises Winsløw's model of the concrete to abstract transition in analysis, is applied to discuss the issue of transitions. Our research results cover a description of structuralist praxeologies developed throughout the Bachelor around the notion of ideal, a discussion of discontinuities and ruptures that are observed, and the identification of strategies developed by teachers to handle these transitions as they carry out the internal didactic transposition.

Complemented subsets and Boolean-valued, partial functions

We study the two algebras of complemented subsets that were introduced in the constructive development of the Daniell approach to measure and integration within Bishop-style constructive mathematics. We present their main properties both for the so-called here categorical complemented subsets and for the extensional complemented subsets. We translate constructively the classical bijection between subsets and Boolean-valued, total functions by establishing a bijection between complemented subsets (categorical or extensional) and Boolean-valued, partial functions (categorical or extensional). The role of Myhill’s axiom of non-choice in the equivalence between categorical and extensional subsets is discussed. We introduce swap algebras of type (I) and (II) as an abstract version of Bishop’s algebras of complemented subsets of type (I) and (II), respectively, and swap rings as an abstract version of the structure of Boolean-valued partial functions on a set. Our examples of swap algebras and swap rings together with the included here results indicate that their theory is a certain generalisation of the theory of Boolean algebras and Boolean rings, a fact which we find interesting both from a constructive and a classical point of view.

Straight domains are locally divided

We present a proof that all straight domains are locally divided—thereby answering two open problems posed by Dobbs and Picavet, which appeared in the survey “Open Problems in Commutative Ring Theory” written by Cahen, Fontana, Frisch, and Glaz. In fact, we are able to prove a stronger result: a prime ideal of a domain is straight if and only if it is locally divided.

N-absorbing ideal factorization in commutative rings

In this article, we show that Mori domains, pseudo-valuation domains, and n-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain R is a Mori locally pseudo-valuation domain if and only if each proper ideal of R is a finite product of 2-absorbing ideals of R. Moreover, every ideal of a Mori locally almost pseudo-valuation domain can be written as a finite product of 3-absorbing ideals. To provide concrete examples of such rings, we study rings of the form A + XB [ X ] where A is a subring of a commutative ring B and X is indeterminate, which is of independent interest, and along with several characterization theorems, we prove that in such a ring, each proper ideal is a finite product of n-absorbing ideals for some n ≥ 2 if and only if A ⊆ B is essentially a finite product of field extensions. A complete description of when an order of a quadratic number field is a locally pseudo valuation domain, a locally almost pseudo valuation domain or a locally conducive domain is given.

Development and implementation of Concept-Test questions in abstract algebra

In tertiary mathematics courses, students often have difficulties acquiring an understanding of the mathematical concepts covered. One approach to address this problem is to implement so-called Concept-Tests. These are multiple-choice questions whose distractors represent common problems and misconceptions related to the concepts. While there exist lots of such questions for calculus, Concept-Test questions focusing on basic concepts of abstract algebra are still rare, although previous research has shown that students have many problems with these. We therefore developed such questions for important concepts of basic group and ring theory in the years 2020–2022. In this paper, we first want to present the questions and the developmental process. Furthermore, we want to present an empirical study investigating to what extent the questions helped students in a proof-oriented abstract algebra course to acquire an understanding of the concepts covered. This study especially indicates that the developed Concept-Test questions provided good starting points for conceptual changes.

CDMO: Chaotic Dwarf Mongoose Optimization Algorithm for feature selection

In this paper, a modified version of Dwarf Mongoose Optimization Algorithm (DMO) for feature selection is proposed. DMO is a novel technique of the swarm intelligence algorithms which mimic the foraging behavior of the Dwarf Mongoose. The developed method, named Chaotic DMO (CDMO), is considered a wrapper-based model which selects optimal features that give higher classification accuracy. To speed up the convergence and increase the effectiveness of DMO, ten chaotic maps were used to modify the key elements of Dwarf Mongoose movement during the optimization process. To evaluate the efficiency of the CDMO, ten different UCI datasets are used and compared against the original DMO and other well-known Meta-heuristic techniques, namely Ant Colony optimization (ACO), Whale optimization algorithm (WOA), Artificial rabbit optimization (ARO), Harris hawk optimization (HHO), Equilibrium optimizer (EO), Ring theory based harmony search (RTHS), Random switching serial gray-whale optimizer (RSGW), Salp swarm algorithm based on particle swarm optimization (SSAPSO), Binary genetic algorithm (BGA), Adaptive switching gray-whale optimizer (ASGW) and Particle Swarm optimization (PSO). The experimental results show that the CDMO gives higher performance than the other methods used in feature selection. High value of accuracy (91.9–100%), sensitivity (77.6–100%), precision (91.8–96.08%), specificity (91.6–100%) and F-Score (90–100%) for all ten UCI datasets are obtained. In addition, the proposed method is further assessed against CEC’2022 benchmarks functions.

ANALISIS KEMAMPUAN PEMBUKTIAN MATEMATIS PADA MATA KULIAH TEORI RING DITINJAU DARI PENDIDIKAN SEKOLAH MENENGAH

Mathematical proof is a crucial component in studying mathematics, especially at advanced levels. Previous research has reported that many students face difficulties in solving mathematical proof problems, including in rings theory course. This study aims to determine the level of students' mathematical proof abilities in ring theory course and to compare these abilities based on their school backgrounds. This research employs a quantitative descriptive approach followed by comparative analysis. The subjects were 30 students from the mathematics education program taking rings theory course at Tidar University. The research instruments include tests, questionnaire, and interviews guidelines. Data analysis of the test results was conducted using descriptive statistic and one-way ANOVA followed by Tukey HSD post hoc tests. The research findings indicate that students' overall mathematical proof abilities fall into the low category, with an average score of 41.67 (on a scale of 100). From these results, 84% of the respondents are in the low category, 13% in the medium category, and 3% in the high category. Furthermore, when considering the backgrounds of secondary schools MA, SMK, and SMA, the average scores for mathematical proof abilities in all three groups are in the low category. The one-way ANOVA and Tukey HSD post hoc tests reveal a significant difference in mathematical proof abilities between the MA and SMA groups, while there was no significant difference between the MA and SMK groups or the SMK and SMA groups

Cognitive Flexibility in Developing Higher-Order Thinking Skills in the Course of Ring Theory

Mata kuliah Teori Ring merupakan salah satu mata kuliah bidang matematika yang cukup ketat dalam memberlakukan sistem deduktif-aksiomatik, namun mahasiswa kesulitan memecahkan masalah karena kurang bisa berpikir secara fleksibel saat menerima informasi baru. Tujuan penelitian ini adalah untuk mengetahui hubungan antara cognitive flexibility dengan keterampilan berpikir tingkat tinggi atau Higher-Order Thinking Skills (HOTS). Metode penelitian yang digunakan adalah korelasional Ex Pose Facto. Penelitian dilakukan di Prodi Pendidikan Matematika UIN Sunan Gunung Djati Bandung dengan jumlah mahasiswa yang diteliti sebanyak 120 orang sebesar 1,2%. Teknik analisis data yang digunakan adalah uji regresi linier sederhana. Berdasarkan hasil penelitian dapat disimpulkan bahwa Cognitive Flexibility memiliki pengaruh positif terhadap kemampuan berpikir tingkat tinggi sebesar . Pengaruh positif ini bermakna semakin meningkatnya Cognitive Flexibility mahasiswa maka akan berpengaruh terhadap peningkatan Kemampuan Berpikir Tingkat Tinggi mahasiswa, sehingga akan memberikan impact pada kemampuan yang dihasilkan dari keterampilan mengerjakan soal HOTS terhadap kemampuan fleksibilitas kognitif yang pada akhirnya terwujud sebuah produk manusia yang siap menghadapi tantangan kompleksitas masalah. The Ring Theory course is one of the mathematics courses that are quite strict in applying deductive-axiomatic systems, but students have difficulty solving problems because they are less able to think flexibly when receiving new information. This study aimed to determine the relationship between cognitive flexibility and Higher-Order Thinking Skills (HOTS). The research method used is correlational Ex Pose Facto. The research was conducted at the Mathematics Education UIN Sunan Gunung Djati Bandung with the number of students studying as many as 120 people amounting to 1.2%. The data analysis technique used is a simple linear regression test. Based on the study results, it can be concluded that Cognitive Flexibility positively influences higher-order thinking skills. This positive influence means that the increasing Cognitive Flexibility of students will affect the improvement of students' higher-order thinking skills, it will have an impact on the abilities resulting from the skills of doing HOTS questions on the ability of cognitive flexibility which ultimately materializes a human product that is ready to face the challenges of problem complexity.

Commutativity of some Prime Nearrings using Left-Sided outer (σ, τ)-n-Derivation

In the field of mathematics, pure mathematics is very important as it gives rise to the basis for the formation of all applicable mathematical concepts in solving real-life problems. Algebra is such an integral part of pure mathematics. It consists of the Ring theory. It has been discovered that there exists some structures similar to rings with little deformity and they are called NEARRINGS. These structures do not commute mostly as they fail to satisfy distributive law. To ascertain the commutativity of nearrings, we need derivation(s). Because several papers had been presented dealing with left-nearrings, this paper aimed to consider right-nearrings which has not been done before in that respect, to the best of our knowledge. Some methods dealing with left-nearrings have been studied and modified in this work, and new derivation has been introduced to take care of right-nearrings. For Let

A novel method for ring-planet gear mesh force identification via SVD-based Kalman filter

Dynamic mesh forces are the primary vibration excitations of the planetary gear transmission systems. However, they are difficult to measure directly, and conventional indirect measurement methods are typically only applicable to specific planetary gear sets with special measuring equipment. This study proposes a more widely applicable method for identifying the mesh forces of ring-planet gear pairs by utilizing vibration signals. To this end, a ring gear dynamic model is established based on the elastic theory of the thick ring in the form of a space state system. The state vector of the model is constructed from parameters of ring gear vibration responses, while the input state vector is derived from parameters of dynamic mesh forces. A singular value decomposition (SVD)-based Kalman filter is employed to provide a stable solution for the estimation of state vector and input vector simultaneously. A mesh force indirect measurement method based on tooth fillet strain signals collected by strain gauges is adopted to evaluate the accuracy of identification results. Results of dynamic experiments conducted in a planetary gear set test rig demonstrate that compared with measured results, the proposed method can obtain acceptable estimated mesh forces of ring-planet gear pairs by using vibration signals.

The impact of human cadaveric dissection on professional identity formation in medical students

BackgroundAs technology advances, some schools are moving away from human cadaveric dissection to teach anatomy, leading to concern regarding the possible loss of a professional identity building experience. This study explored the role of dissection in students’ professional identity formation.MethodsA mixed-methods study was conducted using survey methodology and semi-structured interviews of medical students at an American-style graduate-entry medical school in Singapore. The questionnaire adopted the conceptual framework of the Ring Theory of Personhood and the MacLeod-Clark Professional Identity Scale was used to measure professional identity, followed by semi-structured interviews of students using Braun and Clarke’s six-phase reflexive thematic analysis.ResultsRespondents did not differ substantively from non-respondents by age, nationality, or ethnicity, and year of entering medical school, however, they were slightly more female dominant. The number of hours of hands-on participation in dissection showed no significant relationship (r2 = 0.010; p = 0.424) with professional identity formation measured by the MacLeod-Clark Professional Identity Scale. Despite the survey results, semi-structured interviews revealed rich and nuanced findings suggesting the influence of dissection in participants’ professional identity formation through deepening students’ appreciation of humanistic values and enhancing their notions of patients’ personhood. Notably, students without dissection experience did not express these sentiments and were orientated towards knowledge acquisition.ConclusionWhile our findings do not suggest that dissection strongly impacts students’ professional identity formation, students shared thought-provoking experiences which suggest some level of its contribution. Careful consideration of this phenomenon should be exercised prior to removing dissection in favour of technological alternatives.

Minimal Prime and Semiprime Submodules

Prime and semiprime submodules are important generalizations of prime and semiprime ideals to module theory over commutative rings. However, minimal or smallest prime/semiprime submodules have received comparatively less attention. This paper furthers the theory of minimal prime and minimal semiprime submodules through several new directions. After formally introducing these concepts, we establish characterization theorems for minimal prime submodules based on intersections of primes and associated primes. A complete structure theory is also developed for minimal semiprime submodules, allowing their description up to isomorphism. We introduce prime and semiprime operators which stably preserve primeness properties between modules. Results on the persistence of minimal primeness of submodules under such operators are proved. Connections of our minimal semiprime characterization back to classic ring theory are highlighted, recovering past results on minimal prime ideals. Overall, this paper significantly expands the foundations of minimal prime and semiprime submodules, while opening up many new questions at this nexus of module and ring theory.

Reliability Function and Boolean Ring

Reliability functions are fundamental mathematical models in engineering reliability theory, giving the probability that a system or component functions over a given time period. Boolean rings are algebraic structures with addition, multiplication, and complementation operations that follow certain axioms. This paper shows that collections of reliability functions form Boolean rings under natural definitions of the Boolean operators. Although reliability theory and Boolean rings are mature subjects, the formal connection between them has not been fully recognized. Establishing this link allows the extensive body of results from Boolean ring theory to be applied in analyzing reliability functions. The Boolean perspective leads to simplified calculations and new reliability theorems. We provide mathematical background on reliability functions and Boolean rings before presenting the formal proof that reliability functions satisfy Boolean ring properties. Several examples demonstrate how the Boolean viewpoint yields insight into reliability calculations, system structure functions, and other areas. The relationships revealed here unify reliability theory and Boolean algebra, while offering opportunities for further theoretical developments in both fields. This work elucidates the underlying Boolean structure inherent in reliability modeling.

Research status of phase change memory and its materials

Phase change memory technology is a new technology in non-volatile memory technology. Phase change memory technology has many advantages, such as non-volatility, high reading and writing speeds, better data retention, and strong compatibility with CMOS technology, and has been paid attention to by many researchers. Phase change materials are mainly chalcogenide compound materials. Researchers have done a lot of research on Ge-Sb-Te, Ge-Te, and Sb-Te-based phase change materials and developed some new phase change material systems. In addition, researchers have carried out in-depth research on the phase transition mechanism, but the existing atomic umbrella jump theory, multiple ring theory, resonance bond theory, octahedral structure theory, etc. have not formed a unified understanding of the phase transition mechanism. At present, phase change memory is mainly prepared by magnetron sputtering of phase change material targets, which results in fast deposition and high purity of the prepared film. In this paper, based on the development of phase change storage materials, the system of phase change materials and its phase change mechanism, the phase transition mechanism of phase change memory, the preparation and characterization methods of phase change films, the industrialization progress, and other research work are reviewed.

What is the Digitalised Curriculum for?

The emergence of the deadly corona virus, popularly known as COVID-19, caused a hasty and ill-prepared paradigm shift in higher education institutions. Formerly reliant on the face-to-face mode of teaching and learning, the need for physical distancing recommended by the World Health Organization forced them to adopt a digitalised curriculum. This article exploreswhether the digitalised curriculum adopted by a higher education institution in Lesotho was for qualification, socialisation and/ or subjectification. In other words, it investigates which propositions are influenced by the digitalised curriculum. Purposive sampling was used to select five studies conducted in Lesotho during/post the COVID-19 pandemic. The tree three rings theory was used to frame the study, with thematic analysis through critical discourse analysis employed to analyse the data. The findings suggest that, to some extent, the use of a learning management system favours qualification at the expense of both socialisation and subjectification. In addition, the digital divide was glaringly evident in the adoption of online learning. Formal incorporation of social media sites is recommended to enable students to socialise with prescribed content and to utilise their unique experiences with digital technologies to achieve their prescribed goals.

Assessing the effects of a mentoring program on professional identity formation

BackgroundMedical education has enjoyed mixed fortunes nurturing professional identity formation (PIF), or how medical students think, feel and act as physicians. New data suggests that structured mentoring programs like the Palliative Medicine Initiative (PMI) may offer a means of developing PIF in a consistent manner. To better understand how a well-established structured research mentoring program shapes PIF, a study of the experiences of PMI mentees is proposed.MethodologyAcknowledging PIF as a sociocultural construct, a Constructivist approach and Relativist lens were adopted for this study. In the absence of an effective tool, the Ring Theory of Personhood (RToP) and Krishna-Pisupati Model (KPM) model were used to direct this dual Systematic Evidence-Based Approach (Dual-SEBA) study in designing, employing and analysing semi-structured interviews with PMI mentees and mentoring diaries. These served to capture changes in PIF over the course of the PMI’s mentoring stages.Transcripts of the interviews and mentoring diaries were concurrently analysed using content and thematic analysis. Complementary themes and categories identified from the Split Approach were combined using the Jigsaw Approach and subsequently compared with mentoring diaries in the Funnelling Process. The domains created framed the discussion.ResultsA total of 12 mentee interviews and 17 mentoring diaries were analysed, revealing two domains—PMI as a Community of Practice (CoP) and Identity Formation. The domains confirmed the centrality of a structured CoP capable of facilitating longitudinal mentoring support and supporting the Socialisation Process along the mentoring trajectory whilst cultivating personalised and enduring mentoring relationships.ConclusionThe provision of a consistent mentoring approach and personalised, longitudinal mentoring support guided along the mentoring trajectory by structured mentoring assessments lay the foundations for more effective mentoring programs. The onus must now be on developing assessment tools, such as a KPM-based tool, to guide support and oversight of mentoring relationships.

Some Studies on Commutative Rings in Commutative Algebra

In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of noncommutative ring where multiplication is not required to be commutative.

On Bergman's Diamond Lemma for Ring Theory

This expository and review paper deals with the Diamond Lemma for ring theory, which is proved in the first section of G. M. Bergman, The Diamond Lemma for Ring Theory, Advances in Mathematics, 29 (1978), pp. 178–218. No originality of the present note is claimed on the part of the author, except for some suggestions and figures. Throughout this paper, I shall mostly use Bergman’s expressions in his paper. In Remarks and Notes, the reader will find some useful information on this topic.

On Symmetric Additive Mappings and Their Applications

The key motive of this paper is to study symmetric additive mappings and discuss their applications. The study of these symmetric mappings makes it possible to characterize symmetric n-derivations and describe the structure of the quotient ring S/P, where S is any ring and P is a prime ideal of S. The symmetricity of additive mappings allows us to transfer ring theory results to functional analyses, particularly to C∗-algebras. Precisely, we describe the structures of C∗-algebras via symmetric additive mappings.

Gradings on block-triangular matrix algebras

Upper triangular, and more generally, block-triangular matrices, are rather important in Linear Algebra, and also in Ring theory, namely in the theory of PI algebras (algebras that satisfy polynomial identities). The group gradings on such algebras have been extensively studied during the last decades. In this paper we prove that for any group grading on a block-triangular matrix algebra, over an arbitrary field, the Jacobson radical is a graded (homogeneous) ideal. As noted by F. Yasumura [Arch. Math. (Basel) 110 (2018), pp. 327–332] this yields the classification of the group gradings on these algebras and confirms a conjecture made by A. Valenti and M. Zaicev [Arch. Math. (Basel) 89 (2007), pp. 33–40].