Prime Elements Research Articles

The Macías topology on integral domains

In this manuscript a recent topology on the positive integers generated by the collection of {σn:n∈N} where σn:={m:gcd⁡(n,m)=1} is generalized over integral domains. Some of its topological properties are studied. Properties of this topology on infinite principal ideal domains that are not fields are also explored, and a new topological proof of the infinitude of prime elements is obtained (assuming the set of units is finite or not open), different from those presented in the style of H. Furstenberg. Finally, some problems are proposed.

N-prime elements and the primality of x − α in D⟦x⟧

Let D be an integral domain and D [ ​ [ x ] ​ ] be the power series ring over D. In this paper, we study the primality of x − α in D [ ​ [ x ] ​ ] , where α ∈ D . For this purpose, we generalize the definition of a prime element as follows. For a positive integer N, a nonzero nonunit α ∈ D is called an N-prime element if for any a , b ∈ D , α N | ab implies α | a or α | b . We prove that if α is an N-prime element for some N, then x − α is a prime element in D [ ​ [ x ] ​ ] . Surprisingly, it is shown that the converse also holds when D is a PID, a valuation domain, or a Dedekind domain. In other words, when D is a PID, a valuation domain, or a Dedekind domain, a necessary and sufficient condition for x − α to be a prime element in D [ ​ [ x ] ​ ] is α is an N-prime element in D for some N. This however does not hold for other types of integral domains such as UFDs or Krull domains. We also investigate the N-prime property in an arbitrary integral domain and give other (equivalent) conditions for an element α in the aforementioned types of integral domains to be an N-prime element.

Revisiting Baer elements

The objective of this paper is to extend certain properties observed in $d$-ideals of rings and $d$-elements of frames to Baer elements in multiplicative lattices. Additionally, we present results concerning these elements that have not been addressed in the study of $d$-ideals of rings. Furthermore, we introduce Baer closures and explore Baer maximal, prime, semiprime and meet-irreducible elements.

Lengths of factorizations of integer-valued polynomials on Krull domains with prime elements

Let D be a Krull domain admitting a prime element with finite residue field and let K be its quotient field. We show that for all positive integers k and 1<n1≤⋯≤nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1 < n_1 \\le \\cdots \\le n_k$$\\end{document}, there exists an integer-valued polynomial on D, that is, an element of Int(D)={f∈K[X]∣f(D)⊆D}\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Int}\\,}}(D) = \\{ f \\in K[X] \\mid f(D) \\subseteq D \\}$$\\end{document}, which has precisely k essentially different factorizations into irreducible elements of Int(D)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\,\ extrm{Int}\\,}}(D)$$\\end{document} whose lengths are exactly n1,…,nk\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n_1, \\ldots , n_k$$\\end{document}. Using this, we characterize lengths of factorizations when D is a unique factorization domain and therefore also in case D is a discrete valuation domain. This solves an open problem proposed by Cahen, Fontana, Frisch, and Glaz.

Difference Ordered G􀀀 Semirings

Objectives: In this study, we generalize some of the difference ordered and weak uniquely difference ordered semirings results. Methods: To establish the results in semirings, we use conditions like commutativity, simple, multiplicative cancellative, additively idempotent on difference-ordered, and weak uniquely difference-ordered semirings. Findings: First, we give some examples of difference ordered semiring, and weak difference ordered semirings. Then generalize some of the results of semirings to semirings and discuss some of the properties of additive idempotent semifield. Novelty: We find that if is a non-zeroic difference ordered semiring then is a strong ideal of Let be a positive difference ordered Gel’fand semiring then every maximal non-unit of is prime. Further, we find that if is a simple difference ordered additively idempotent semiring and which is not a unit then is prime if and only if there exists a character : satisfying Moreover, if and are distinct prime elements of Then and are also distinct. Finally, we consider some properties of additive idempotent semifield and then introduce the concept of weak uniquely difference-ordered semirings. AMS Mathematics subject classification (2020): 16Y60. Keywords: Γ-semiring, Difference ordered Γ-semiring, Additively idempotent Γ- semiring, Strong identity, Weak uniquely difference-ordered Γ-semirings

Prime Spectrum Graph of C-Lattices

Let £ be a C-lattice. Let σ(£) be the set of all prime elements of £ and M(£) be the collection of all maximal elements of £. For S ⊆M(£), we introduce the new graph called S-join graph on σ(£), denoted by ΓS(σ(£)). We have studied properties like connectivity, diameter and domination number of the graph σ(£) . In this paper, we established that the topological space σ(£) is connected if and only if the graph ΓS(σ(£)) is connected.

Rings and radicals related to n-primariness

This paper concerns ring properties which are induced from the structure of the powers of prime ideals. An ideal [Formula: see text] of a ring [Formula: see text] is called [Formula: see text]-primary (respectively, [Formula: see text]-primary) provided that [Formula: see text] for ideals [Formula: see text] of [Formula: see text] implies that [Formula: see text] or [Formula: see text] is nil of index [Formula: see text] (respectively, [Formula: see text] or [Formula: see text] is nil) in [Formula: see text], where [Formula: see text]. It is proved that for a proper ideal [Formula: see text] of a principal ideal domain [Formula: see text], [Formula: see text] is [Formula: see text]-primary if and only if [Formula: see text] is of the form [Formula: see text] for some prime element [Formula: see text] and [Formula: see text] if and only if [Formula: see text] is [Formula: see text]-primary, through which we study the structure of matrices over principal ideal domains. We prove that for a [Formula: see text]-primary ideal [Formula: see text] of a ring [Formula: see text], [Formula: see text] is prime when the Wedderburn radical of [Formula: see text] is zero. In addition we provide a method of constructing strictly descending chain of [Formula: see text]-primary radicals from any domain, where the [Formula: see text]-primary radical of a ring [Formula: see text] means the intersection of all the [Formula: see text]-primary ideals of [Formula: see text].

ROLE OF PANCHAKARMA IN GYNAECOLOGICAL DISEASES

Artava, Shonita, Asrika, Raja, Rakta, Lohita, Rudhir and Pushpa are some of the words used in the Ayurvedic classics to denote menstrual blood or ovum. Some Acharaya consider Artava as Updhatu of Rasa while others consider it as Updhatu of Rakta. Artava is Agneya and has characteristics similar to Rakta. According to Ayurvedic science, Artava, in various forms, maintains women's reproductive health. When Artava becomes vitiated, it can become the prime cause of diseases in women. Hence, Artava is considered a prime element in maintaining women's health. As the normal function of Artava can maintain the balance in the female reproductive system, the same balance can get disturbed when there is vitiation of Dosha, manifesting conditions like Artava Dusti and Yonivyapada. Panchakarma can play a vital role in treating these abnormal conditions.

Minimization of hypersurfaces

Let F ∈ Z [ x 0 , … , x n ] F \in \mathbb {Z}[x_0, \ldots , x_n] be homogeneous of degree d d and assume that F F is not a ‘nullform’, i.e., there is an invariant I I of forms of degree d d in n + 1 n+1 variables such that I ( F ) ≠ 0 I(F) \neq 0 . Equivalently, F F is semistable in the sense of Geometric Invariant Theory. Minimizing F F at a prime p p means to produce T ∈ M a t ( n + 1 , Z ) ∩ G L ( n + 1 , Q ) T \in Mat(n+1, \mathbb {Z}) \cap GL(n+1, \mathbb {Q}) and e ∈ Z ≥ 0 e \in \mathbb {Z}_{\ge 0} such that F 1 = p − e F ( [ x 0 , … , x n ] ⋅ T ) F_1 = p^{-e} F([x_0, \ldots , x_n] \cdot T) has integral coefficients and v p ( I ( F 1 ) ) v_p(I(F_1)) is minimal among all such F 1 F_1 . Following Kollár [Electron. Res. Announc. Amer. Math. Soc. 3 (1997), pp. 17–27], the minimization process can be described in terms of applying weight vectors w ∈ Z ≥ 0 n + 1 w \in \mathbb {Z}_{\ge 0}^{n+1} to F F . We show that for any dimension n n and degree d d , there is a complete set of weight vectors consisting of [ 0 , w 1 , w 2 , … , w n ] [0,w_1,w_2,\dots ,w_n] with 0 ≤ w 1 ≤ w 2 ≤ ⋯ ≤ w n ≤ 2 n d n − 1 0 \le w_1 \le w_2 \le \dots \le w_n \le 2 n d^{n-1} . When n = 2 n = 2 , we improve the bound to d d . This answers a question raised by Kollár. These results are valid in a more general context, replacing Z \mathbb {Z} and p p by a PID R R and a prime element of R R . Based on this result and a further study of the minimization process in the planar case n = 2 n = 2 , we devise an efficient minimization algorithm for ternary forms (equivalently, plane curves) of arbitrary degree d d . We also describe a similar algorithm that allows to minimize (and reduce) cubic surfaces. These algorithms are available in the computer algebra system Magma.

On prime elements in commutative domains

On prime elements in commutative domains

Characterization of Fatigue Crack Growth Behaviour of Alloy 617M Base and Weld Material

Alloy 617M is a Ni-based super-alloy made in the form of tubes which are used in high-temperature boilers and super heaters in Advanced Ultra Super Critical (AUSC) power plants. The high-temperature oxidation and mechanical properties of this material have been derived based on chromium (Cr), cobalt (Co) and molybdenum (Mo) as prime alloying elements which cause solid solution strengthening, carbide precipitation (M23C6, M6C) and formation of intermetallic phase Ni3(Al, Ti) (γʹ-phase). The manufacturing route of these components involve bending and welding operations which are critical locations and potential sites for crack nucleation and growth. Also, the conjoint influence of high temperature and flow-induced vibrations in the tubes further aides to crack nucleation and growth. Thus, fatigue crack growth (FCG) properties are important. Therefore, in this work, detailed experimental studies on the FCG behaviour of Alloy 617M base and weld materials have been carried out at different test temperatures (27 °C, 650 °C, 710 °C and 750 °C). The effect of ageing (1000, 2000, 5000 and 20000 hours) at an operating temperature of 710 °C on FCG has also been evaluated. All the FCG tests have been conducted as per ASTM with a constant load ratio (R = 0.1) and frequency of 15 Hz. From the experimental results, a major trend is observed of decreasing threshold stress intensity factor (ΔKTH) with an increase in ageing time and test temperature. The variations in the ΔKTH with test temperature and ageing time have been discussed and the crack growth mechanism has been established through microstructure and fractographic studies validating from existing literatures.

Prefab Facades – From Prototype to Product?

Building envelopes are not only the prime element of the exterior aesthetic quality of buildings; they have also become a major driver both for building construction cost and operational performance. The importance of prefabrication is growing in the building industry as it allows faster, high-quality, and cost-effective construction while reducing risks associated with onsite labour. Although prefabrication for structural components is a relatively recent development, it has been widely used in the manufacturing of building envelopes for many years, particularly in the case of unitized curtain wall systems. However, whether using prefabricated components or not, façade design development remains a challenge due to the need for façade engineers to rapidly develop technically viable and financially feasible solutions that achieve the desired architectural design intent. Particularly at the early stages of the design process, the turnaround for multiple iterations is often fast-paced, and abortive work is, therefore, not uncommon. This paper outlines an approach to addressing this challenge, attempting to bridge the gap between façade design, fabrication, and installation. A new design approach and tools are presented that allow designers to iteratively validate concepts based on a pre-engineered system that is optimized for performance and take fabrication, transport, installation costs, maintenance, and circularity into account. As a result, the tool/design workflow will ensure consistent quality, meeting budgets and timelines while enhancing material efficiency and fostering energy-conscious and circular envelope design approaches.

Two topologies on the set of minimal prime elements of a continuous frame

In this paper, we discuss the properties of the hull-kernel topology and the inverse topology on the set [Formula: see text] of minimal prime elements of a continuous frame [Formula: see text]. We prove that the set [Formula: see text] endowed with the hull-kernel topology and the inverse topology, respectively, is a [Formula: see text]-space. Moreover, we obtain that there is a bijective correspondence between the set [Formula: see text] of all maximal Scott open filters of [Formula: see text] and the set [Formula: see text] when [Formula: see text] is a stably continuous frame. By considering the bijective correspondence between the sets [Formula: see text] and [Formula: see text], we propose some sufficient conditions for the topological spaces [Formula: see text] endowed with the hull-kernel topology and the inverse topology, respectively, to be sober, Hausdorff, compact, extremely disconnected and zero-dimensional spaces, respectively, when [Formula: see text] is a stably continuous frame.

Efficient Prediction of Seasonal Infectious Diseases Using Hybrid Machine Learning Algorithms with Feature Selection Techniques

Early seasonal disease risk identification is the most challenging task in the medical industry. The capacity to detect patients at risk of improvement throughout their hospital stay is critical for effective patient allocation and care among patients with seasonal diseases. Patient risk factor prediction is the most important issue in reducing victim mortality. Machine learning plays a vital role in identifying the risk level of patients. In this research, we consider four seasonal diseases such as dengue, malaria, typhoid, and pneumonia. In this research, the researcher finds the dangerous symptoms of victims based on their age group. The real-time dataset was used in this study. The dataset for this study was gathered from hospitals in the Madurai district between 2019 and 2020. Feature selection is the prime element of patient risk recognition. It is used to pick the most accurate attributes for prediction. The dataset is divided into 70% training data and 30% testing data. This research proposes the feature selection method Boruta-XGBoost for improving accuracy. In this research, we discuss various attribute selection algorithms, including the Boruta algorithm, the XGBoost algorithm, the recursive feature elimination method (RFE), and the PRF-BXGBoost (Patient Risk Factor-Boruta-XGBoost) algorithm. The proposed method provides greater accuracy when compared to other variable selection methods. The time complexity of the proposed method is low when compared to other algorithms.

Infection Control in Dentistry: An Evaluation

The dental environment is associated with significant amount of risk for exposure to various micro- organisms. Infection control is one of the prime elements of a successful dental practice. There are many infectious diseases that can be transmitted in a dental environment. New diseases with serious consequences and a high rate of transmission have evolved in the recent past. Infection control is directed at prevention to exposure of such infections and also to prevent it being transferred from person to person. Purpose: The goal of an infection control program is to provide a safe working environment for dental health care personnel and their patients. Practitioners can achieve this by adopting measures that reduce health care-associated infections among patients and occupational exposures among dental health care personnel. It is crucial for all dental practitioners to be up to date on current Centers for Disease Control and Prevention guidelines, equipment, and techniques for proper infection control. Continuous evaluation of infection control practices is important. Patients and dental providers should be confident that oral health care can be delivered and received in a safe manner. Summary: Most steps in infection control routine are directed at prevention of contact with infectious agent. Personal protection barriers are of great significance in this process. Various methods like sterilization, decontamination and disinfection are indispensable. The WHO and CDC have issued certain guidelines regarding prevention and dealing with certain infective conditions. Dentists need to update themselves about these guidelines, to be able to administer these policies effectively. Use of disposable items may be expensive but is an effective and simple means of infection control. More and more disposable items are being inducted into dental practices especially critical items like needles, syringe etc. With careful evaluation and analysis, infection control routine can be employed in daily practice with the resources available. Constant monitoring and evaluation of infection control routines is required for good effectiveness.

On the cancellation problem for L-algebras

In this paper, we consider the cancellation problem for the cartesian product of [Formula: see text]-algebras. Firstly, we show that [Formula: see text]-algebras with prime element [Formula: see text] and [Formula: see text]-algebras satisfying the condition ([Formula: see text]) are cancellable. Furthermore, we also prove that the wedge-sum of cancellable [Formula: see text]-algebras is cancellable and each [Formula: see text]-algebra can be embedded into a cancellable [Formula: see text]-algebra. Finally, we give a class of [Formula: see text]-algebras satisfying the cancellation law which is different from the above [Formula: see text]-algebras.

An Efficient Method to Obtain Initial Basic Feasible Solution of a Transportation Problem

Transportation Problem (TP) is the study of strategy for shipping the product from the origin (supplier) to the destination (consumers) and the optimal shipment cost of the TP obtained from the initial basic feasible solution (IBFS). IBFS is the prime element of TP which helps to optimize the TP objective function. Existing methods for finding IBFS could not give the nearest value to minimal cost everywhere. Hence an innovative technique Known as TOCM-MT provides IBFS, which is the closest to an optimal solution. Many researchers have used this TOCM-MT technique through thousands of real-life TP problems, in every time got a better result than VAM, JHM, and TDM1 techniques. Hence, this technique is best for obtaining IBFS. This research is illustrated the TOCM-MT approach through time minimizing, transshipment, and profit maximization problems with some numerical examples.

BaTiO3:Sm3+ nanophosphor with K+ ion incorporation for modulating non-radiative transitions

Rare earth (RE) activated nanophosphors are the prime elements employed to manufacture light emitting diodes (LEDs) for the current solid state lighting (SSL) industry. The apparent lack of reddish orange emitting nanophosphors is proving to a constraint in the commercialization of the white light emitting diodes (WLEDs). Herein, the size of BaTiO3 (BTO): Sm3+ and K+ co-activated BTO: Sm3+ nanophosphor, with an average particle size of 80 nm, have been produced by a modified sol gel technique. The synthesized nanophosphors emit a brilliant reddish-orange light when excited at 406 nm. The relative photoluminescence (PL) studies of Sm3+ doped BTO and Sm3+ doped BTO with K+ nanophosphor show that adding K+ doubles the intensity of the emitted light and improves the thermal stability in a significant way. The results of the research indicated that using the aforementioned nanophosphor in the future may be advantageous for solid-state lighting systems, including warm LEDs with cyan light chips.

Two kinds of enriched topological representations of Q-algebras

Abstract In this paper, we continue the study of the enriched topological representation of $Q$-algebras where $Q$ is a unital quantale and give two kinds of enriched topological representations of $Q$-algebras. The first one is based on strong $M_3$-valued $Q$-algebra homomorphisms, and the second way is based on strong $M_6$-valued $Q$-algebra homomorphisms. For the first way, we first construct a spatial and semiunital $Q$-algebra $M_3$ containing three elements and show that prime elements of semiunital $Q$-algebras are identified with strong $Q$-algebra homomorphisms taking their values in $M_3$. Then we prove that a semiunital $Q$-algebra is spatial iff strong $Q$-algebra homomorphisms with values in $M_3$ separate elements. Based on this, we obtain that every spatial and semiunital $Q$-algebra can be identified with an $M_3$-enriched sober space. For the second enriched topological representation, we construct a spatial and semiunital $Q$-algebra $M_6$ containing exactly six elements and prove that every spatial and semiunital $Q$-algebra can be identified with an $M_6$-enriched sober space.

Current Molecular, Cellular and Genetic Aspects of Peri-Implantitis Disease: A Narrative Review

(1) Background: Peri-implantitis is a multi-factorial disease with an inflammatory background that occurs in both soft and hard tissues surrounding implants. In recent years, the understanding of the cellular, molecular and genetic background of peri-implantitis has broadened. This study aims to summarize the currently available articles on the subject and highlight the most recent advances over the last 20 years. (2) Methods: For this study, the Embase and PubMed libraries were searched using the keywords: ("peri-implantitis" AND "cytokine" OR "genetics" OR "cellular") and ("peri-implantitis" AND "cytokine" OR "genetics" OR "cellular" AND "risk factors"). The search revealed a total of 3013 articles (992 from PubMed, 2021 from Embase). Following screening of the titles and abstracts and full-text reads, 55 articles were included. (3) Results: In peri-implantitis IL-6, IL-1β, TNF-α, MMP-8 and their genetic variations appear to be the most important cytokines in relation to not only pathogenesis, but also their potential diagnostic capabilities. Epithelial and inflammatory cells, along with those of the bone lineage, are prime cellular elements found in peri-implantitis. (4) Conclusions: A wide array of cells stand behind peri-implantitis, as well as cytokines and their genetic variations that take part in the process. However, the growing interest in this topic has led to the introduction of specific new diagnostic tools to enable a better understanding of patients' responses to treatment and, in turn, to even enable prediction of the risk of developing peri-implant disease.