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Mock‐Lie algebras, a unique class of commutative algebras that satisfy the Jacobi identity, are gaining attention for their potential applications in various mathematical contexts. This study presents a matrix representation of the centroid for Mock‐Lie algebras in dimensions up to four, offering insights into their structure and interactions. By applying the centroid concept and utilizing matrix operations, we explore the relationships between matrix elements, leading to a better understanding of these algebras. Our findings aim to contribute to the growing interest in Mock‐Lie algebras and their role within the broader algebraic landscape. [ABSTRACT FROM AUTHOR]

An element a of a ring R is called a quasi-idempotent if a 2 = k a for some central unit k of R , or equivalently, a = k e , where k is a central unit and e is an idempotent of R. A ring R is called a quasi-Boolean ring if every element of R is quasi-idempotent. A ring R is called (strongly) quasi-clean if each of its elements is a sum of a quasi-idempotent and a unit (that commute). These rings are shown to be a natural generalization of the clean rings and strongly clean rings. An extensive study of (strongly) quasi-clean rings is conducted. The abundant examples of (strongly) quasi-clean rings state that the class of (strongly) quasi-clean rings is very larger than the class of (strongly) clean rings. We prove that an indecomposable commutative semilocal ring is quasi-clean if and only if it is local or R has no image isomorphic to ℤ 2 ; For an indecomposable commutative semilocal ring R with at least two maximal ideals, n (R) (n ≥ 2) is strongly quasi-clean if and only if n (R) is quasi-clean if and only if min { | R / m | , m is a maximal ideal of R } > n + 1. For a prime p and a positive integer n ≥ 2 , n (ℤ (p)) is strongly quasi-clean if and only if p > n. Some open questions are also posed. [ABSTRACT FROM AUTHOR]

Let ℚ be the rational filed. For a square-free integer d with d ≠ 0 , 1 , we denote by K = ℚ (d) the quadratic field. Let K be the ring of algebraic integers of K. In this paper, we completely determine the unit group of the quotient ring K / 〈 ξ n 〉 of K for an arbitrary prime ξ in ℚ (d) , where ℚ (d) has the unique factorization property, and n > 1 is a rational integer. [ABSTRACT FROM AUTHOR]

The power graph P G of a finite group G is the graph whose vertex set is G, two distinct vertices are adjacent if one is a power of the other. The order supergraph S G of P G is the graph with vertex set G, and two distinct vertices x, y are adjacent if o(x)|o(y) or o(y)|o(x). In this paper, we study the independence number of S G and answer a question was posed by Hamzeh and Ashrafi. [ABSTRACT FROM AUTHOR]

Let R be a ring with non-zero identity. The unit graph of R , denoted by G (R) , is an undirected graph with all the elements of R as vertices and where distinct vertices x , y are adjacent if and only if x + y is a unit of R. In this paper, we investigate the Wiener index and hyper-Wiener index of G (R) and explicitly determine their values. [ABSTRACT FROM AUTHOR]

In the era of the rise of big data, if insurance companies can effectively and reasonably use existing data to tap more potential information, it can not only improve work efficiency but also achieve precise marketing to customers, thereby saving costs. At the same time, China pays more and more attention to financial insurance, so it is of great significance to use customer information to explore the purchase behavior of financial insurance. This paper mainly analyzes the factors that affect the purchase of financial insurance from the aspects of individuals and families and provides references for insurance marketing. This paper selects the results of the China Comprehensive Social Survey in 2020 as the empirical sample data of this paper and sets whether to purchase financial insurance as the target variable. The characteristic variables were screened, and the 10 variables selected after screening were finally used to build the model. Next, the data are divided into training set and test set, and a decision tree learning model is established on the two data sets at the same time. The classification results of the model are evaluated according to the classification evaluation indicators. Finally, the decision tree analysis results provide suggestions and strategies for the construction of insurance marketing recommendation methods. [ABSTRACT FROM AUTHOR]

Athletes must maintain their physical fitness in order to compete in any sport. The event can be organized for a single person or multiple people. Irrespective of the number of people or team participation in a sport, the people should have perfect training. The performance and physical fitness of the candidate will be measured under various categories, and the data will be stored in the database. The data to be collected about each event, player, coach, and others will result in the creation of big data with the aid of artificial intelligence and wireless networking. Wireless networking aids in the collection of data around the globe in a shorter period with the aid of intelligent servers. In this study, a recursive Bayesian estimation algorithm is implemented to perform the analysis of training and testing of the athlete's performance with physical training. The proposed algorithm achieved an accuracy of 99%, which is a minimum increase in nine (09) percentage points over the neural network and an 18% growth over the fuzzy set model. The proposed models are able to analyze players' success at a higher level based on their scores at each factor level. The experimental results show that the proposed model outperforms well in enhancing player performance. [ABSTRACT FROM AUTHOR]

In the process of tobacco silk making, how to better improve the quality of stem and leaf separation has become an issue of concern. His research mainly discusses the data collected from tobacco leaf images based on computer vision sensors. In LM (Levenberg-Marquarelt) as a training function, the algorithm uses threshing effect samples for training and learning. This paper is aimed at extracting the shape characteristic parameters of tobacco leaves and obtains the shape parameters of the length, width, area, circumference, and roundness of the tobacco leaves. In this paper, boundary tracking is used to obtain the coordinate and direction information of the tobacco leaf boundary pixels in the image, which provides a basis for obtaining the subsequent extraction of tobacco leaf characteristic parameters. In the tobacco leaf grading system, the tobacco leaf feature parameter extraction module displays the geometric characteristics of tobacco leaf, such as length, width, area, aspect ratio, rectangularity, and color characteristic, hue H , saturation S , A channel, and B channel in detail through the computer vision sensor. Finally, the subjective and objective combination weighting method is used to combine and weight the indicators of the threshing effect of the first-level threshing machine, which not only considers the quantity of information provided by the indicators but also takes into account the subjective view of the experts, which increases the weight of the indicators, accuracy, and scientificity. The approximation accuracy of the training samples of the threshing effect prediction model based on the BP neural network LM algorithm is 99.495%, the approximation accuracy of the validation set is 96.535%, and the approximation accuracy of the test set is 98.392%. This research will greatly improve production efficiency and meet the enterprise's requirements for high efficiency and low cost. [ABSTRACT FROM AUTHOR]

A ring is said to satisfy the strong 2-sum property if every element is a sum of two commuting units. In this note, we present some sufficient or necessary conditions for the matrix ring over a commutative local ring to have the strong 2-sum property. [ABSTRACT FROM AUTHOR]

The unit graph of a ring R is the simple graph, denoted by G (R) , whose vertex set is R , and in which two distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this paper, we completely determine the domination number of G (ℤ n) when n has exactly three prime divisors. [ABSTRACT FROM AUTHOR]

We investigate the formal matrix ring over R defined by a certain system of factors. We give a method for constructing formal matrix rings from non-negative integer matrices. We also show that the principal factor matrix of a binary system of factors determine the structure of the system. [ABSTRACT FROM AUTHOR]

The unit graph of a ring is the simple graph whose vertices are the elements of the ring and where two distinct vertices are adjacent if and only if their sum is a unit of the ring. A simple graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this note, we completely characterize the semipotent rings whose unit graphs are planar. As a consequence, we list all semilocal rings with planar unit graphs. [ABSTRACT FROM AUTHOR]

The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. Let R be an associative (not necessarily commutative) ring. We focus on the domination number of the zero-divisor graph Γ (R) , the compressed zero-divisor graph Γ E (R) and the unit graph G (R). We find some relations between the domination number of the zero-divisor graph and that of the compressed zero-divisor graph. Moreover, some relations between the domination number of Γ (R) and Γ (R [ x ; α , δ ]) , as well as the relations between the domination number of G (R) and G (R [ [ x ; α ] ]) , are studied. [ABSTRACT FROM AUTHOR]

Motivated by Hirano-Tominaga's work on rings for which every element is a sum of two idempotents and by de Seguins Pazzis's results on decomposing every matrix over a field of positive characteristic as a sum of idempotent matrices, we address decomposing every matrix over a commutative ring as a sum of three idempotent matrices and, respectively, as a sum of three involutive matrices. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity. The unitary Cayley graph of R is the simple graph with vertex set R , where two distinct vertices x and y are linked by an edge if and only if x − y is a unit of R. A graph is said to be planar if it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In this paper, we completely characterize the rings whose unitary Cayley graphs are planar. [ABSTRACT FROM AUTHOR]

For a square-free integer d other than 0 and 1, let $$K = \mathbb{Q}\left( {\sqrt d } \right)$$, where $$\mathbb{Q}$$ is the set of rational numbers. Then K is called a quadratic field and it has degree 2 over $$\mathbb{Q}$$. For several quadratic fields $$K = \mathbb{Q}\left( {\sqrt d } \right)$$, the ring R of integers of K is not a unique-factorization domain. For d < 0, there exist only a finite number of complex quadratic fields, whose ring R of integers, called complex quadratic ring, is a unique-factorization domain, i.e., d = −1,−2,−3,−7,−11,−19,−43,−67,−163. Let ϑ denote a prime element of R, and let n be an arbitrary positive integer. The unit groups of R/< ϑ> was determined by Cross in 1983 for the case d = −1. This paper completely determined the unit groups of R/< ϑ> for the cases d = −2,−3. [ABSTRACT FROM AUTHOR]

Ring extensions are a well-studied topic in ring theory. In this paper, we study the structure of the Gauss extension of a Galois ring. We determine the structures of the extension ring and its unit group. [ABSTRACT FROM AUTHOR]

Let R be a ring with identity. The unit graph of R, denoted by G(R), is a simple graph with vertex set R, and where two distinct vertices x and y are adjacent if and only if x + y is a unit in R. The genus of a simple graph G is the smallest nonnegative integer g such that G can be embedded into an orientable surface Sg. In this paper, we determine all isomorphism classes of finite commutative rings whose unit graphs have genus at most three. [ABSTRACT FROM AUTHOR]

The unit graph of a ring R, denoted G(R), is the simple graph defined on the elements of R with an edge between vertices x and y iff x + y is a unit of R. In this paper, we prove that the girth (G(R)) of the unit graph of an arbitrary ring R is 3, 4, 6 or ∞. We determine the rings R with R/J(R) semipotent and with (G(R)) = 6 or ∞, and classify the rings R with R/J(R) right self-injective and with (G(R)) = 3 or 4. [ABSTRACT FROM AUTHOR]

For a finite commutative ring R, the square mapping graph of R is a directed graph Γ(R) whose set of vertices is all the elements of R and for which there is a directed edge from a to b if and only if a2=b. We establish necessary and sufficient conditions for the existence of isolated fixed points, and the cycles with length greater than 1 in Γ(R). We also examine when the induced subgraph on the set of zero-divisors of a local ring with odd characteristic is semiregular. Moreover, we completely determine the finite commutative rings whose square mapping graphs have exactly two, three or four components. [ABSTRACT FROM AUTHOR]