• The Mathematical Education
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• v.42 no.3
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• pp.265-274
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• 2003

To learn number system is a basic thing in school mathematics. In this paper, we analyze how number system is introduced in the 7th national curriculum and text books according to it. We found that the method of introducing number system is quite different from the former curriculum and former text books. There are many negative effects on the transition when compared with those of United States of America. There are many gaps between elementary school mathematics and middle school mathematics about levels and skills to deal with numbers. It is desirable to distinguish the negative sign from subtraction and to keep the concepts of commutativity and associativity of operations from the beginning. We also suggest to change the introducing timing and extension methods of number system in 7th grade mathematics.

• Journal of the Korean Mathematical Society
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• v.49 no.4
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• pp.855-865
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• 2012

Recently, we have introduced a group invariant, which is related to the number of elements $x$ and $y$ of a finite group $G$ such that $x{\wedge}y=1_{G{\wedge}G}$ in the exterior square $G{\wedge}G$ of $G$. This number gives restrictions on the Schur multiplier of $G$ and, consequently, large classes of groups can be described. In the present paper we generalize the previous investigations on the topic, focusing on the number of elements of the form $h^m{\wedge}k$ of $H{\wedge}K$ such that $h^m{\wedge}k=1_{H{\wedge}K}$, where $m{\geq}1$ and $H$ and $K$ are arbitrary subgroups of $G$.

• Communications of the Korean Mathematical Society
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• v.34 no.1
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• pp.127-136
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• 2019

This article concerns the normal property of elements on Jacobson and nil radicals which are generalizations of commutativity. A ring is said to be right njr if it satisfies the normal property on the Jacobson radical. Similarly a ring is said to be right nunr (resp., right nlnr) if it satisfies the normal property on the upper (resp., lower) nilradical. We investigate the relations between right duo property and the normality on Jacobson (nil) radicals. Related examples are investigated in the procedure of studying the structures of right njr, nunr, and nlnr rings.

• Bulletin of the Korean Mathematical Society
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• v.58 no.2
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• pp.327-347
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• 2021

As an extension to the study of Toeplitz operators on the Bergman space, the notion of H-Toeplitz operators B�� is introduced and studied. Necessary and sufficient conditions under which H-Toeplitz operators become co-isometry and partial isometry are obtained. Some of the invariant subspaces and kernels of H-Toeplitz operators are studied. We have obtained the conditions for the compactness and Fredholmness for H-Toeplitz operators. In particular, it has been shown that a non-zero H-Toeplitz operator can not be a Fredholm operator on the Bergman space. Moreover, we have also discussed the necessary and sufficient conditions for commutativity of H-Toeplitz operators.

• Communications of the Korean Mathematical Society
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• v.37 no.4
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• pp.1147-1170
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• 2022

In this paper, we introduce a new class of mappings called the generalized orthogonal F-Suzuki contraction for a family of multivalued mappings in the setup of orthogonal b-metric spaces. We established the existence of some common fixed point results without using any commutativity condition for this new class of mappings in orthogonal b-metric spaces. Moreover, we illustrate and support these common fixed point results with example. The results obtained in this work generalize and extend some recent and classical related results in the existing literature.

• Journal of Applied and Pure Mathematics
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• v.6 no.3_4
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• pp.167-176
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• 2024

This study is about division and right multiplication in matrices. The discussion of the properties of multiplication and division is examined. Some results between multiplication based on the row-column relationship and division based on the same relationship are discussed. The commonalities of these results between the processes are emphasized. Examples of unrealized properties are given. The algebraic properties of the newly defined right product and division are clarified in matrices. The properties of the known multiplication operation and new situations between right multiplication and division are investigated. Some results are declared between the transpositions of matrices and the obtained rules of operations. New results are discussed belong the equations ${\underleftarrow{XA}}=B$ and ${\underleftarrow{AX}}=B$. New ideas are proposed for solving these equations. The contribution The contribution is explained the equation $AB={\underleftarrow{BA}}$ to division operation. Many new properties, lemmas and theorems are presented on this subject.

• Communications of the Korean Mathematical Society
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• v.39 no.3
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• pp.585-593
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• 2024

Let 𝒦 be a ring. An additive map 𝖚^⋄ → 𝖚 is called Jordan involution on 𝒦 if (𝖚^⋄)^⋄ = 𝖚 and (𝖚𝖛+𝖛𝖚)^⋄ = 𝖚^⋄𝖛^⋄+𝖛^⋄𝖚^⋄ for all 𝖚, 𝖛 ∈ 𝒦. If Θ is a (non-zero) 𝜂-generalized derivation on 𝒦 associated with a derivation Ω on 𝒦, then it is shown that Θ(𝖚) = 𝛄𝖚 for all u ∈ 𝒦 such that 𝛄 ∈ Ξ and 𝛄² = 1, whenever Θ possesses [Θ(𝖚), Θ(𝖚^⋄)] = [𝖚, 𝖚^⋄] for all 𝖚 ∈ 𝒦.

• Kyungpook Mathematical Journal
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• v.59 no.1
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• pp.1-11
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• 2019

Let R be a prime ring with involution of the second kind and centre Z(R). Suppose R admits a generalized derivation $F:R{\rightarrow}R$ associated with a derivation $d:R{\rightarrow}R$. The purpose of this paper is to study the commutativity of a prime ring R satisfying any one of the following identities: (i) $F(x){\circ}x^*{\in}Z(R)$ (ii) $F([x,x^*]){\pm}x{\circ}x^*{\in}Z(R)$ (iii) $F(x{\circ}x^*){\pm}[x,x^*]{\in}Z(R)$ (iv) $F(x){\circ}d(x^*){\pm}x{\circ}x^*{\in}Z(R)$ (v) $[F(x),d(x^*)]{\pm}x{\circ}x^*{\in}Z(R)$ (vi) $F(x){\pm}x{\circ}x^*{\in}Z(R)$ (vii) $F(x){\pm}[x,x^*]{\in}Z(R)$ (viii) $[F(x),x^*]{\mp}F(x){\circ}x^*{\in}Z(R)$ (ix) $F(x{\circ}x^*){\in}Z(R)$ for all $x{\in}R$.

• Journal of Educational Research in Mathematics
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• v.21 no.3
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• pp.239-259
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• 2011

This study investigated the elementary school students' ability on the algebraic reasoning as generalized arithmetic. It analyzed the written responses from 648 second graders, 688 fourth graders, and 751 sixth graders using tests probing their understanding of the properties of whole number operations. The result of this study showed that many students did not recognize the properties of operations in the problem situations, and had difficulties in applying such properties to solve the problems. Even lower graders were quite successful in using the commutative law both in addition and subtraction. However they had difficulties in using the associative and the distributive law. These difficulties remained even for upper graders. As for the associative and the distributive law, students had more difficulties in solving the problems dealing with specific numbers than those of arbitrary numbers. Given these results, this paper includes issues and implications on how to foster early algebraic reasoning ability in the elementary school.

• Journal of Elementary Mathematics Education in Korea
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• v.21 no.4
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• pp.643-662
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• 2017

The first appearance of the equations in elementary school mathematics is in the expression of the equal sign in the addition sentences without its definition. Most elementary school students have operational understanding of the equal sign in equations. Moreover, students' opportunities to have a clear concept of the properties of operations are limited because they are used implicitly in the textbooks. Based on this fact, it has been argued that it is necessary to introduce the properties of operations explicitly in terms of specific numbers and to deal with various types of equations for understanding a relational meaning of the equal sign. In this study, we use equations to represent the implicit properties of operations and the relational meaning of the equal sign in elementary school mathematics with respect to students' level of understanding. In addition, we give some explicit examples which show how to apply them to make efficient computations.