• Kyungpook Mathematical Journal
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• 제59권3호
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• pp.391-401
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• 2019

A module is said to be ${\pi}$-extending provided that every projection invariant submodule is essential in a direct summand of the module. In this article, we focus on the class of modules having the ${\pi}$-extending property by looking at the singularity of quotient submodules. By doing so, we provide counterexamples, using hypersurfaces in projective spaces over complex numbers, to show that being generalized ${\pi}$-extending is not inherited by direct summands. Moreover, it is shown that the direct sums of generalized ${\pi}$-extending modules are generalized ${\pi}$-extending.

• Korean Journal of Mathematics
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• 제31권1호
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• pp.1-16
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• 2023

In this paper, we examine the relations of two closely related concepts, the digital Lusternik-Schnirelmann category and the digital higher topological complexity, with each other in digital images. For some certain digital images, we introduce κ-topological groups in the digital topological manner for having stronger ideas about the digital higher topological complexity. Our aim is to improve the understanding of the digital higher topological complexity. We present examples and counterexamples for κ-topological groups.

• Kyungpook Mathematical Journal
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• 제63권4호
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• pp.521-537
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• 2023

A module M is called a simple-direct-injective module if, whenever A and B are simple submodules of M with A ≅ B and B is a direct summand of M, then A is a direct summand of M. Some new characterizations of these modules are proved. The structure of simple-direct-injective modules over a commutative Dedekind domain is fully determined. Also, some relevant counterexamples are indicated to show that a left simple-direct-injective ring need not be right simple-direct-injective.

• 대한수학회논문집
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• 제39권1호
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• pp.71-77
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• 2024

Let A be a commutative integral domain with identity element and S a multiplicatively closed subset of A. In this paper, we introduce the concept of S-valuation domains as follows. The ring A is said to be an S-valuation domain if for every two ideals I and J of A, there exists s ∈ S such that either sI ⊆ J or sJ ⊆ I. We investigate some basic properties of S-valuation domains. Many examples and counterexamples are provided.

• 논리연구
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• 제13권1호
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• pp.21-52
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• 2010

반프라센의 숙고 원리는 반례에 직면한다는 주장들이 있는데 이 중 두드러지는 것이 잠자는 미녀 문제가 그런 반례라는 주장이다. 엘가는 잠자는 미녀 문제에 대한 자신의 대답이 숙고 원리의 반례를 보여준다고 주장한다. 반면 잠자는 미녀 문제에 대한 루이스의 대답은 숙고 원리를 보존한다. 최근 보스트롬은 숙고의 원리를 유지하면서 엘가의 입장과 루이스의 입장을 절충하려는 혼성 이론을 제시한 바 있다. 그는 엘가의 입장이 '자기 표시 가정'을 전제하고 있으며 이 가정은 옳지 못하다고 주장한다. 하지만 나는 엘가가 보스트롬의 행위자-부분 개념을 이용하면 오히려 보스트롬의 비판을 피할 수 있다고 본다. 또한 잠자는 미녀 문제를 포함하여 몇 가지 확률 관련 퍼즐들을 살펴봄으로써 미래 자아의 견해를 다루는 올바른 입장을 제시한다. 이 새롭게 제안된 입장에 따르면, 행위자의 한 명제에 대한 믿음의 정도에서 중요한 것은, 그가 외부자 입장과 내부자 입장 중 어떤 것을 취하는가 하는 것이다.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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• pp.1363-1379
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• 2011

Halpern iterative algorithm is one of the most cited in the literature of approximation of fixed points of nonexpansive mappings. Different authors modified this iterative algorithm in Banach spaces to approximate fixed points of nonexpansive mappings. One of which is Hu [8] and Yao et al [21] modification of Halpern iterative algorithm for nonexpansive mappings in Banach spaces. It is the purpose of this paper to thoroughly analyze this modification and its convergence conditions. Unfortunately, Hu [8] and Yao et al [21] control conditions imposed on the modified Halpern iterative algorithm to have strong convergence are found to be not sufficient. In this paper, counterexamples are constructed to prove that the strong convergence conditions of Hu [8] and Yao et al [21] are not sufficient. It is also proved that with some additional conditions on the control parameters, strong convergence of the defined iterative algorithm is obtained in different Banach space settings.

• 대한수학회보
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• 제36권3호
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• pp.579-588
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• 1999

In this paper we study the connections between right quasi-duo rings and 2-primal rings, including several counterexamples for answers to some questions that occur naturally in the process. Actually we concern following three questions and modified ones: (1) Are right quasi-duo rings 2-primal$\ulcorner$, (2) Are formal power series rings over weakly right duo rings also weakly right duo\ulcorner and (3) Are 2-primal rings right quasi-duo\ulcorner Moreover we consider some conditions under which the answers of them may be affirmative, obtaining several results which are related to the questions.

• 대한수학회지
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• 제47권4호
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• pp.719-733
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• 2010

We observe a structure on the products of coefficients of nilpotent polynomials, introducing the concept of n-semi-Armendariz that is a generalization of Armendariz rings. We first obtain a classification of reduced rings, proving that a ring R is reduced if and only if the n by n upper triangular matrix ring over R is n-semi-Armendariz. It is shown that n-semi-Armendariz rings need not be (n+1)-semi-Armendariz and vice versa. We prove that a ring R is n-semi-Armendariz if and only if so is the polynomial ring over R. We next study interesting properties and useful examples of n-semi-Armendariz rings, constructing various kinds of counterexamples in the process.

• 대한수학회논문집
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• 제22권2호
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• pp.207-208
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• 2007

In this work, authors considered a result concerning elliptic curves $y^2=x^3+cx$ over $\mathbb{F}_p$ mod 8, given at [1]. They noticed that there should be a slight change at this result. They give counterexamples and the correct version of the result.

• 한국수학교육학회지시리즈B:순수및응용수학
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• 제2권2호
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• pp.127-132
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• 1995

[1] Show that ∫$\_$0/$\^$1/ [∫$\_$0/$\^$1/ f($\chi$,y)dy] d$\chi$ = ∫$\_$0/$\^$1/[∫$\_$0/$\^$1/ f($\chi$,y)d$\chi$] Counterexample: If pk denotes the k-th prime number, let S(pk) = (equation omitted), let S = ∪$\_$k=1/$\^$$\infty$/ S(pk), and let Q = [0, 1]${\times}$[0, 1]. Define f on Q as follows; f($\chi$, y) = 0 ($\chi$, y)$\in$S, f($\chi$, y) = 1 ($\chi$, y)$\in$Q - S.(omitted)