• The Pure and Applied Mathematics
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• v.12 no.3 s.29
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• pp.193-209
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• 2005

We introduce the notions of intuitionistic fuzzy prime ideals, intuitionistic fuzzy completely prime ideals and intuitionistic fuzzy weakly completely prime ideals. And we give a characterization of intuitionistic fuzzy ideals and establish relationships between intuitionistic fuzzy completely prime ideals and intuitionistic fuzzy weakly completely prime ideals.

• East Asian mathematical journal
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• v.18 no.1
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• pp.15-20
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• 2002

We study the relationships between strongly prime ideals and completely prime ideals, concentrating on the connections among various radicals(prime radical, upper nilradical and generalized nilradical). Given a ring R, consider the condition: (*) nilpotent elements of R form an ideal in R. We show that a ring R satisfies (*) if and only if every minimal strongly prime ideal of R is completely prime if and only if the upper nilradical coincides with the generalized nilradical in R.

• International Journal of Fuzzy Logic and Intelligent Systems
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• v.12 no.3
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• pp.198-204
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• 2012

We introduce the notions of interval-valued fuzzy prime ideals, interval-valued fuzzy completely prime ideals and interval-valued fuzzy weakly completely prime ideals. And we give a characterization of interval-valued fuzzy ideals and establish relationships between interval-valued fuzzy completely prime ideals and interval-valued fuzzy weakly completely prime ideals.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.3
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• pp.459-466
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• 2009

The purpose of this paper is to study the some properties of fuzzy quasi-semiprime ideal, fuzzy prime ideals and to prove some fundamental properties of semigroups. In particular, we will establish a relation between fuzzy prime ideals and weakly completely semiprime ideals by using the some equivalent conditions of fuzzy semiprime ideals.

• Bulletin of the Korean Mathematical Society
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• v.44 no.1
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• pp.151-156
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• 2007

A connection between weak ${\pi}-regularity$ and the condition every prime ideal is maximal will be investigated. We prove that a certain 2-primal ring R is weakly ${\pi}-regular$ if and only if every prime ideal is maximal. This result extends several known results nontrivially. Moreover a characterization of minimal prime ideals is also considered.

• Communications of the Korean Mathematical Society
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• v.27 no.3
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• pp.483-488
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• 2012

In this paper we introduce the notion of P-strongly regular near-ring. We have shown that a zero-symmetric near-ring N is P-strongly regular if and only if N is P-regular and P is a completely semiprime ideal. We have also shown that in a P-strongly regular near-ring N, the following holds: (i) $Na$ + P is an ideal of N for any $a{\in}N$. (ii) Every P-prime ideal of N containing P is maximal. (iii) Every ideal I of N fulfills I + P = $I^2$ + P.

• Bulletin of the Korean Mathematical Society
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• v.46 no.6
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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).

• Communications of the Korean Mathematical Society
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• v.24 no.2
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• pp.161-169
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• 2009

In this paper, we introduce the generalized ideal-based zero-divisor graph structure of near-ring N, denoted by $\widehat{{\Gamma}_I(N)}$. It is shown that if I is a completely reflexive ideal of N, then every two vertices in $\widehat{{\Gamma}_I(N)}$ are connected by a path of length at most 3, and if $\widehat{{\Gamma}_I(N)}$ contains a cycle, then the core K of $\widehat{{\Gamma}_I(N)}$ is a union of triangles and rectangles. We have shown that if $\widehat{{\Gamma}_I(N)}$ is a bipartite graph for a completely semiprime ideal I of N, then N has two prime ideals whose intersection is I.

• Kyungpook Mathematical Journal
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• v.46 no.2
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• pp.261-271
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• 2006

In this paper, we investigate and characterize the class of A-semirings. A characterization of the Thierrin radical of a proper ideal of an A-semiring is given. Moreover, when P is a Q-ideal in the semiring R, it is shown that P is primary if and only if R/P is nilpotent.