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The objective of this research paper is to justify that an additive mapping F from a semiprime ring R to itself will be ϕ-centralizer having a suitable torsion restriction on R if it satisfies certain algebraic equations. [ABSTRACT FROM AUTHOR]

Let $ \mathcal{A} $ be a $ (p+q)! $-torsion free semiprime ring. We proved that if $ \mathcal{H}, \mathcal{D} : \mathcal{A}\to \mathcal{A} $ are two additive mappings fulfilling the algebraic identity $ 2\mathcal{H}(a^{p+q}) = \mathcal{H}(a^p) a^q+ a^p \mathcal{D}(a^q)+\mathcal{H}(a^q) a^p+ a^q \mathcal{D}(a^p) $ for all $ a\in \mathcal{A} $, then $ \mathcal{H} $ is a generalized derivation with $ \mathcal{D} $ as an associated derivation on $ \mathcal{A} $. In addition to that, it is also proved in this article that $ \mathcal{H}_1 $ is a generalized left derivation associated with a left derivation $ \delta $ on $ \mathcal{A} $ if they fulfilled the algebraic identity $ 2\mathcal{H}_1(a^{p+q}) = a^p \mathcal{H}_1(a^q)+ a^q \delta(a^p)+a^q \mathcal{H}_1(a^p)+ a^p \delta(a^q) $ for all $ a \in \mathcal{A} $. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented.

Let A be a (p + q)!-torsion free semiprime ring. We proved that if H, D : A → A are two additive mappings fulfilling the algebraic identity 2H(ap+q) = H(ap)aq + apD(aq) + H(aq)ap + aqD(ap) for all a e A, then H is a generalized derivation with D as an associated derivation on A. In addition to that, it is also proved in this article that H1 is a generalized left derivation associated with a left derivation δ on A if they fulfilled the algebraic identity 2H1(ap+q) = apH1(aq) + aqδ(ap) + aqH1(ap) + apδ(aq) for all a ∈ A. Further, the legitimacy of these hypotheses is eventually demonstrated by examples and at last, an application of Banach algebra is presented. [ABSTRACT FROM AUTHOR]

It is known that mathematics has an important effect on the development of science and technology. Identities, which are widely used in many branches of science, form the basis of some mathematics courses such as algebra. However, it appears as one of the learning areas of secondary school 7th grade mathematics and is seen as one of the difficult subjects to learn. The fact that there are many and abstract formulas about identities is difficult for students to remember. Therefore, visualization is important in teaching abstract concepts in a meaningful way and in associating concepts with daily life. Visualization is the use of mathematical and geometric symbols, shapes, diagrams, etc. to make something unseen imaginable. It is the drawing or showing of the shape and picture of an abstract concept with the help of pencil and computer. Visualization and visual proof are central to mathematics. The visualization approach is used in mathematics and mathematics education as a tool, not as a goal in the lessons. Identities are seen as one of the learning areas that we encounter in the solution of problems that we encounter in our daily lives and that are difficult to learn. In this direction, the visual proofs of the identities in R2 are shown in detail to the researchers thanks to the visualization approach, so that the algebraic identities can be learned more easily. Although identities are an important learning area in mathematics and algebra lessons, they are thought to be difficult to understand and learn by students. In line with this idea, students have difficulty in understanding identities as the visual models from which identities are obtained are not adequately expressed by the teachers and are not associated with daily life. For this reason, it is suggested by mathematics teachers that as a solution to this problem, visualizing abstract mathematical concepts in lessons or showing algebraic and visual proofs of some theorems may be beneficial. In this study, which was planned as a theoretical study, Within the scope of descriptive scanning model, document analysis technique was used. Visual proofs of basic algebraic identities are shown to researchers in detail by using geometric figures. In line with this purpose, some suggestions have been made to shed light on this issue for interested researchers. [ABSTRACT FROM AUTHOR]

In mature symbolic algebra, from Viète onward, the handling of several algebraic unknowns was routine. Before Luca Pacioli, on the other hand, the simultaneous manipulation of three algebraic unknowns was absent from European algebra and the use of two unknowns so infrequent that it has rarely been observed and never analyzed. The present paper analyzes the five occurrences of two algebraic unknowns in Fibonacci's writings; the gradual unfolding of the idea in Antonio de' Mazzinghi's Fioretti; the distorted use in an anonymous Florentine algebra from ca 1400; the regular appearance in the treatises of Benedetto da Firenze; and finally what little we find in Pacioli's Perugia manuscript and in his Summa. It asks which of these appearances of the technique can be counted as independent rediscoveries of an idea present since long in Sanskrit and Arabic mathematics - metaphorically, to which extent they represent reinvention of the hot water already available on the cooker in the neighbour's kitchen; and it raises the question why the technique once it had been discovered was not cultivated - pointing to the line diagrams used by Fibonacci as a technique that was as efficient as rhetorical algebra handling two unknowns and much less cumbersome, at least until symbolic algebra developed, and as long as the most demanding problems with which algebra was confronted remained the traditional recreational challenges. [ABSTRACT FROM AUTHOR]

We continue our analysis of Brahmagupta's BrÀhmasphuÇasiddhÀnta (India, 628), that had shown that each of his sequences of propositions should be read as an apodictic discourse: a connected discourse that develops the natural consequences of explicitly stated assumptions, within a particular conceptual framework. As a consequence, we established that Brahmagupta did provide a derivation of his results on the cyclic quadrilateral. We analyze here, on the basis of the same principles, further problematic passages in Brahmagupta's magnum opus, regarding number theory and algebra. They make no sense as sets of rules. They become clear as soon as one reads them as an apodictic discourse, so carefully composed that they leave little room for interpretation. In particular, we show that (i) Brahmagupta indicated the principle of the derivation of the solution of linear congruences (the kuÇÇaka) at the end of chapter 12 and (ii) his algebra in several variables is the result of the extension of operations on numbers to new types of quantities - negative numbers, surds and "non-manifest" variables. [ABSTRACT FROM AUTHOR]

Abstract: A ()-tensor is introduced in an invariant form. Algebraic identities are derived that connect the Schouten ()-tensor and tensor with the Nijenhuis tensor . Applications to the bi-Hamiltonian dynamical systems are presented. [Copyright &y& Elsevier]

Abstract: The general algebraic identities are discovered for the Nijenhuis and Haantjes tensors on an arbitrary manifold . For , the special algebraic identities involving the symmetric bilinear form are derived. [Copyright &y& Elsevier]