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On prime ring R with a commutator [x, y]=xy - yx, ∀x, y E R and symmetric biderivation B can be constructed symmetric generalized biderivation Δ with associated symmetric biderivation B. However, a prime ring with symmetric generalized biderivation Δ does not always become a commutative ring. This article constructs the conditions for two elements in the prime ring which will be calculated by the product of the commutator so that the ring becomes commutative, i.e there is a non-zero ideal I of R such that one of the following three conditions holds for all x, y, z ∈ I: (1). [Δ(x, y), Δ(y, z)]=0, (2). [Δ(x, y), Δ(y, z)]=[x, z] or (3). [Δ(x, y), Δ(y, z)]+[x, z]=0. So, we get the commutativity of prime rings with generalized symmetric biderivations. [ABSTRACT FROM AUTHOR]

Let R be a unital ring and m ≥ 2 be a positive integer. An element a ∈ R is called an m-potent element in R, if it satisfies am = a. A unital ring R is called m-clean if each of its elements can be written as the sum of an m-potent element and a unit element of R. This article aims to show the existence of m-clean elements in a certain upper triangular matrix ring over integral domain ℤ, by identifying m-potent and unit elements in the ring. So, m-clean elements can be constructed. In this article, we obtained two general forms of all m-potent and unit elements and therefore produced four general forms of all m-clean elements which generalizes some theorems of clean elements in a certain upper triangular matrix ring over integral domain ℤ. Furthermore, we provided some properties of m-cleanness under ring homomorphism. [ABSTRACT FROM AUTHOR]

In algebraic theory, there are groups, rings, modules over a ring, and algebra over rings. As a result, people try to develop theories that apply to groups to apply to other concepts. For example, Goursat's theorem in groups related to the direct product of the two groups, then developed in the ring, the module over a ring, the algebra over a ring related to the direct product of the two rings, two modules over a ring, and two algebras over a ring, respectively. Can we extend Goursat's theorem in algebras over a ring R (R-algebras) with respect to the direct product of n R-algebras? So, this paper will extend Goursat's theorem to direct products of n R-algebras, exploring related properties and providing formal proof. The main result of this study demonstrates that every subalgebra in a direct product of n R-algebras can be uniquely determined by n − 1 R-algebra epimorphisms of an algebra to a factor algebra. [ABSTRACT FROM AUTHOR]

A decomposition of a graph G is a collection of edge-disjoint subgraphs H1, H2, ..., Hr of G such that every edge of G belongs to exactly one Hi. In 2020, Hamiltonian decomposition of Cayley graphs in the dihedral-2p group, where p is a single prime have been studied. In this paper, we study the Hamiltonian decomposition of a Cayley graph of the dihedral-2n group with n ≥ 3. We determine the Hamilton decomposition of the Cayley graphs of a dihedral group on the certain generator {r, r−1, s} of a dihedral group. We show that the Cayley graphs can be decomposed into a Hamiltonian cycle and perfect matching. [ABSTRACT FROM AUTHOR]

The purpose of this study is describing how to solve mathematical problems for elementary students on cube and block material in terms of Adversity Quotient (AQ). This type of research is descriptive qualitative research. The subjects of this research were 35 students of grade 5 MI Nurul Huda Gresik which was then selected two students from each type of AQ. The instruments used are AQ questionnaire, problem solving questions, and interview guidelines. Data analysis techniques used are data reduction, data presentation, and drawing conclusions. The result of the research is that Climber 1 can solve cube and block problems well. CL1 can understand problems, make and implement resolution plan, and recheck the results of his work. Climber 2 can solve cube and block problems well. CL2 can also fulfill indicators of each stage of problem solving well. Camper 1 could not fulfill all stages of problem solving well on cube and block problems. CP1 can understand the problem, make and implement resolutions plan quite well. However, CP1 made a few mistakes in writing formula for the surface area of a cube and slightly made calculation error in finding the length of a block. CP1 also did not recheck the results of his work. Camper 2 can find the correct answer to cube and block problems. CP2 can understand problems and make plans quite well. CP2 in the final plan did not write formula for the volume of a cube. While in finding the size of a block, CP2 does not write down the calculation process. CP2 also did not recheck the results of his work. Quitter 1 and 2 on cube and block problems were only able to fulfill indicators at the stage of understanding the problem. QT1 and QT2 are unable to make and execute plans properly, QT1 and QT2 also did not recheck the results. [ABSTRACT FROM AUTHOR]

From any element a of a ring R with unity, we set τ(a) = {e ∈ R|e3 = e and a - e E Nil(R)}, where Nil(R) is the set of all nilpotent elements of R; trinil clean index of R is defined by sup {|τ(a)||a ∈ R} and it is denoted by Trinin(R). Motivated by the trinil clean index of any ring Zn for any positive integer n ≥ 2, we expand these results by determined the trinil clean index of upper triangular matrix 2 × 2 over Zn which n is positive integer greater than 2. [ABSTRACT FROM AUTHOR]

An element x in a ring R with unity is called almost clean if there exists some idempotent element e in R and a regular element u in R such that x = e + u. This article aims to show the existence of almost clean elements in a certain matrix ring over an integral domain R with unity and provide its characterizations. [ABSTRACT FROM AUTHOR]

The purpose of this study is to analyze the algebraic thinking process of SMA Negeri 1 Bojonegoro students in solving absolute value mathematics problem. The type of this research is qualitative descriptive. Based on report card grades, prospective subjects are classified into 3 ability categories: high thinking ability, medium thinking ability, and low thinking ability. The result obtained are students with high thinking ability are able to write and elaborate the mathematical models, apply the properties of inequality absolute values, and able to get the correct answer. Students with medium thinking ability are able to understand the problems contents, write and elaborate the mathematical models, and working carefully, but unable to make a different representation. Students with low thinking ability are able to understand the problems contents, and able to write and elaborate the mathematical models, but unable to develop a mathematical model that has been made correctly, wrong in executing the numbers in absolute sign, and wrong in making different representations. [ABSTRACT FROM AUTHOR]

Assessments can be used to document and collect information about learning. This documentation and information can be used to analyze the course of learning and materials for self-reflection, for teachers, and for students. This research involved 15 students of the Mathematics Study Program, Faculty of Mathematics and Natural Sciences, Universitas Negeri Malang. This study aims to see the effect of journal writing assessment on the improvement of students' computational thinking skills. This research uses a qualitative approach. The results showed the affective/attitudinal aspect. Students stated that patterns recognition and abstraction was the most difficult part. Decomposition is the easiest part. The analytical assessment shows the average score for decomposition component 2.6 of 3, pattern recognition 2.7 of 3, abstraction 1.6 of 2, algorithm 1.9 of 2, and debugging 1.7 of 2. This shows that journal writing helps improve students' computational thinking skills in solving the problem. [ABSTRACT FROM AUTHOR]

We write notation F → (G, H) for graphs F, G and H to mean that if there is any two-colouring, say red and blue, of all edges of F, then the red subgraph contains a copy of G or the blue subgraph contains a copy of H. The graph F is Ramsey (G, H)-minimal if F → (G, H) but F − e ↛ (G, H) for any e ∈ E(F). The class of all Ramsey (G, H)-minimal graphs will be denoted by ℜ(G, H). In this paper, we prove that there is only one graph that has 6 vertices and 9 edges in ℜ(P3, C6) and we determine some graphs in ℜ(P3, C6) [ABSTRACT FROM AUTHOR]

A ring R with unity is called clean if every element x ∈ R can be written as a sum of an idempotent and a unit element in ring R. Meanwhile, a ring with unity is called feebly clean, if every element r in the ring can be written as r = u + e − f, with ef = fe = 0, where u is a unit element in the ring and e, f is an idempotent element in the ring. [7] shows the existence of clean elements in a subring X3(D) of the matrix ring 3 × 3 over an integral domain D. Based on these results, we provide the set of all clean elements in the matrix ring X3(ℤ), we show the existence of feebly clean elements in the matrix ring X3(ℤ), and show the connections between those elements and feebly clean elements in the matrix ring X3(ℤ). These connections are different from [10]. [ABSTRACT FROM AUTHOR]

In this information era, we always interact with the data needed to make a decision. The data information obtained is often not only observed, processed, and displayed but also interpreted. Analytical thinking necessary in interpreting or understanding data and information so that someone can take the right decision. This article aims to describe students' analytical thinking in solving contextual problems. The design of this research is qualitative-descriptive. The participant of this research were four students of VIII grader SMPN 5 Malang, and they were taken based on the results of the student's work on the answer sheet and the results of the interviews. Subjects criteria selected were from students with different strategies in solving contextual problems. The results show that some students use different strategies and some students use more than one strategy that indicates the process of analytical thinking which includes differentiating, organizing, and attributing in finding the solutions of a contextual problem. [ABSTRACT FROM AUTHOR]

In this article, we consider to obtain new anti fuzzy graph from given anti fuzzy graphs. Product of anti fuzzy graphs is an operation on anti fuzzy graphs that produce new anti fuzzy graph. We consider tensor, normal, modular anti fuzzy graphs products, which are adapted from fuzzy graphs products. In general, product of any two anti fuzzy graphs is an anti fuzzy graph. Product of any two strong anti fuzzy graphs is an anti fuzzy graph strong. Normal product of two anti fuzzy graph complete is a complete anti fuzzy graph. Other than that, different of anti fuzzy products are also discussed. [ABSTRACT FROM AUTHOR]

Edge coloring in a graph is called a rainbow connected if each pair of graph vertices has a rainbow path (i.e., a path with distinct edge colors). The fewest colors utilized so that each pair of graph vertices has a rainbow path is called a rainbow connection number. Meanwhile, if each pair of graph vertices has the shortest path with no edges of the same color, this graph can be called strongly rainbow connected. This path is best known as a rainbow geodesic. The strong rainbow connection number is the fewest colors utilized so that every two vertices in a graph have a rainbow geodesic. In this paper, we determine a strong rainbow connection number of sandat graphs as well as a rainbow and strong rainbow connection numbers of graphs that are obtained from the corona product between a sandat graph St(n) and the complement complete graph K n ¯ . For the results, we obtained that a strong rainbow connection number of St (n) = n with n > 1, and a rainbow and strong rainbow connection number on the corona product of sandat graphs with n > 1, equals the number of pendant edges. [ABSTRACT FROM AUTHOR]