• 대한수학회지
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• 제47권5호
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• pp.1097-1106
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• 2010

Let R be a commutative ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will investigate some ring theoretic properties of R by considering $\Gamma$(R), the zero-divisor graph of R, under the regular action on X by G as follows: (1) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then there is a vertex of $\Gamma$(R) which is adjacent to every other vertex in $\Gamma$(R) if and only if R is a local ring or $R\;{\simeq}\;\mathbb{Z}_2\;{\times}\;F$ where F is a field; (2) If R is a local ring such that X is a union of n distinct orbits under the regular action of G on X, then all ideals of R consist of {{0}, J, $J^2$, $\ldots$, $J^n$, R} where J is the Jacobson radical of R; (3) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then the number of all ideals is finite and is greater than equal to the number of orbits.

• 대한수학회보
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• 제56권3호
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• pp.729-743
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• 2019

In this paper, we introduce strongly quasi primary ideals which is an intermediate class of primary ideals and quasi primary ideals. Let R be a commutative ring with nonzero identity and Q a proper ideal of R. Then Q is called strongly quasi primary if $ab{\in}Q$ for $a,b{\in}R$ implies either $a^2{\in}Q$ or $b^n{\in}Q$ ($a^n{\in}Q$ or $b^2{\in}Q$) for some $n{\in}{\mathbb{N}}$. We give many properties of strongly quasi primary ideals and investigate the relations between strongly quasi primary ideals and other classical ideals such as primary, 2-prime and quasi primary ideals. Among other results, we give a characterization of divided rings in terms of strongly quasi primary ideals. Also, we construct a subgraph of ideal based zero divisor graph ${\Gamma}_I(R)$ and denote it by ${\Gamma}^*_I(R)$, where I is an ideal of R. We investigate the relations between ${\Gamma}^*_I(R)$ and ${\Gamma}_I(R)$. Further, we use strongly quasi primary ideals and ${\Gamma}^*_I(R)$ to characterize von Neumann regular rings.

• 한국수학교육학회지시리즈A:수학교육
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• 제48권1호
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• pp.81-91
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• 2009

This paper focuses on two problems in the 10th grade mathematics, the rational zero theorem and the content(the integer divisor) of a polynomial Among 138 students participated in the problem solving, 58 of them (42 %) has used the rational zero theorem for the factorization of polynomials. However, 30 of 58 students (52 %) consider the rational zero theorem is a mathematical fake(false statement) and they only use it to get a correct answer. There are three different types in the textbooks in dealing with the content of a polynomial with integer coefficients. Computing the greatest common divisor of polynomials, some textbooks consider the content of polynomials, some do not and others suggest both methods. This also makes students confused. We suggests that a separate section of the rational zero theorem must be included in the text. As for the content of a polynomial, we consider the polynomials are contained in the polynomial ring over the rational numbers. So computing the gcd of polynomials, guide the students to give a monic(or primitive) polynomial as ail answer.

• Kyungpook Mathematical Journal
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• 제56권4호
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• pp.1103-1113
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• 2016

In this paper, we study a directed simple graph ${\Gamma}_S(N)$ for a near-ring N, where the set $V^*(N)$ of vertices is the set of all left N-subsets of N with nonzero left annihilators and for any two distinct vertices $I,J{\in}V^*(N)$, I is adjacent to J if and only if IJ = 0. Here, we deal with the diameter, girth and coloring of the graph ${\Gamma}_S(N)$. Moreover, we prove a sufficient condition for occurrence of a regular element of the near-ring N in the left annihilator of some vertex in the strong zero-divisor graph ${\Gamma}_S(N)$.

• 대한수학회지
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• 제48권2호
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• pp.301-309
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• 2011

Let R = $Mat_2(F)$ be the ring of all 2 by 2 matrices over a finite field F, X the set of all nonzero, nonunits of R and G the group of all units of R. After investigating some properties of orbits under the left (and right) regular action on X by G, we show that the graph automorphisms group of $\Gamma(R)$ (the zero-divisor graph of R) is isomorphic to the symmetric group $S_{|F|+1}$ of degree |F|+1.

• 대한수학회보
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• 제46권6호
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• pp.1051-1060
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• 2009

In this paper, we give topological properties of collection of prime ideals in 2-primal near-rings. We show that Spec(N), the spectrum of prime ideals, is a compact space, and Max(N), the maximal ideals of N, forms a compact $T_1$-subspace. We also study the zero-divisor graph $\Gamma_I$(R) with respect to the completely semiprime ideal I of N. We show that ${\Gamma}_{\mathbb{P}}$ (R), where $\mathbb{P}$ is a prime radical of N, is a connected graph with diameter less than or equal to 3. We characterize all cycles in the graph ${\Gamma}_{\mathbb{P}}$ (R).

• Kyungpook Mathematical Journal
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• 제60권4호
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• pp.723-729
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• 2020

Let ℤ_n be the ring of integers modulo n and let ℤ_n[X]] be either ℤ_n[X] or ℤ_n[[X]]. Let 𝚪(Z_n[X]]) be the zero-divisor graph of ℤ_n[X]]. In this paper, we study some properties of 𝚪(ℤ_n[X]]). More precisely, we completely characterize the diameter and the girth of 𝚪(ℤ_n[X]]). We also calculate the chromatic number of 𝚪(ℤ_n[X]]).

• 대한수학회보
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• 제30권1호
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• pp.71-77
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• 1993

Every ring considered in the paper will be assumed to be commutative and have a unit element. An ideal A of a ring R will be called primal if the elements of R which are zero divisors modulo A, form an ideal of R, say pp. If A is a primal ideal of R, P is called the adjoint ideal of A. The adjoint ideal of a primal ideal is prime [2]. The definition of primal ideals may also be formulated as follows: An ideal A of a ring R is primal if in the residue class ring R/A the zero divisors form an ideal of R/A. If Q is a primary idel of a ring R then every zero divisor of R/Q is nilpotent; therefore, Q is a primal ideal of R. That a primal ideal need not be primary, is shown by an example in [2]. Let R[X], and R[[X]] denote the polynomial ring and formal power series ring in an indeterminate X over a ring R, respectively. Let S be a multiplicative system in a ring R and S$^{-1}$ R the quotient ring of R. Let Q be a P-primary ideal of a ring R. Then Q[X] is a P[X]-primary ideal of R[X], and S$^{-1}$ Q is a S$^{-1}$ P-primary ideal of a ring S$^{-1}$ R if S.cap.P=.phi., and Q[[X]] is a P[[X]]-primary ideal of R[[X]] if R is Noetherian [1]. We search for analogous results when primary ideals are replaced with primal ideals. To show an ideal A of a ring R to be primal, it sufficies to show that a-b is a zero divisor modulo A whenever a and b are zero divisors modulo A.

• 대한수학회보
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• 제52권1호
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• pp.247-261
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• 2015

We study the connections between idempotents and zero-divisors in several kinds of ring theoretic properties. We next study several ring theoretic properties and examples related to reversible rings.

• 호남수학학술지
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• 제40권1호
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• pp.75-82
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• 2018

Let P be a commutator poset with Z(P) its set of zero-divisors. The annihilator graph of P, denoted by AG(P), is the (undirected) graph with all elements of $Z(P){\setminus}\{0\}$ as vertices, and distinct vertices x, y are adjacent if and only if $ann(xy)\;{\neq}\;(x)\;{\cup}\;ann(y)$. In this paper, we study basic properties of AG(P).