• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.385-402
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• 2015

In this paper we study the (good) semi-prime closure operations on ideals of a BCK-algebra, lower BCK-semilattice, Noetherian BCK-algebra and meet quotient ideal and then we give several theorems that make different (good) semi-prime closure operations. Moreover by given some examples we show that the given different notions are independent together, for instance there is a semi-prime closure operation, which is not a good semi-prime. Finally by given the notion of "$c_f$-Max X", we prove that every member of "$c_f$-Max X" is a prime ideal. Also we conclude some more related results.

• 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).

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1069-1078
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• 2021

Let R be a commutative ring with identity. A proper ideal I of R is called 1-absorbing primary ([4]) if for all nonunit a, b, c ∈ R such that abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. The concept of 1-absorbing primary ideals in a polynomial ring, in a PID and in idealization of a module is studied. Moreover, we introduce weakly 1-absorbing primary ideals which are generalization of weakly prime ideals and 1-absorbing primary ideals. A proper ideal I of R is called weakly 1-absorbing primary if for all nonunit a, b, c ∈ R such that 0 ≠ abc ∈ I, then either ab ∈ I or c ∈ ${\sqrt{1}}$. Some properties of weakly 1-absorbing primary ideals are investigated. For instance, weakly 1-absorbing primary ideals in decomposable rings are characterized. Among other things, it is proved that if I is a weakly 1-absorbing primary ideal of a ring R and 0 ≠ I₁I₂I₃ ⊆ I for some ideals I₁, I₂, I₃ of R such that I is free triple-zero with respect to I₁I₂I₃, then I₁I₂ ⊆ I or I₃ ⊆ I.

• Korean Journal of Mathematics
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• v.17 no.1
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• pp.77-81
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• 2009

We introduce the notions of a left(right) ${\Gamma}$-filter in a po-${\Gamma}$-semigroups and give a characterization of a left(right) ${\Gamma}$-filter of a po-${\Gamma}$-semigroups in term of right(left) prime ${\Gamma}$-ideals.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.21-28
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• 2015

Let R be a prime ring with center Z, I a nonzero ideal of R, and ${\sigma}$ a non-trivial automorphism of R such that $\{(x{\circ}y)^{\sigma}-(x{\circ}y)\}^n{\in}Z$ for all $x,y{\in}I$. If either char(R) > n or char (R) = 0, then R satisfies $s_4$, the standard identity in 4 variables.

• Journal of applied mathematics & informatics
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• v.39 no.5_6
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• pp.761-768
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• 2021

Let I be a nonzero ideal of a prime ring R and m, n are positive integers. If R admits a generalized derivation F satisfying F(x)ⁿF(y)^m - xⁿy^m = 0 for any y, x ∈ I, then F(x) = ax for any x ∈ R and a^m+n = 1, where a ∈ C, the extended centroid of R.

• Communications of the Korean Mathematical Society
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• v.38 no.1
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• pp.79-87
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• 2023

This paper considers some functional identities related to derivations of a ring R and their action on the centre of R/P where P is a prime ideal of R. It generalizes some previous results that are in the same spirit. Finally, examples proving that our restrictions cannot be relaxed are given.

• Kyungpook Mathematical Journal
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• v.63 no.4
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• pp.551-560
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• 2023

In this paper we consider a more general class of centralizers called I-centralizers. More precisely, given a prime ideal P of an arbitrary ring R we establish a connection between certain algebraic identities involving a pair of P-left centralizers and the structure of the factor ring R/P.

• The Mathematical Education
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• v.12 no.2
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• pp.5-12
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• 1974

단위원을 가지는 하환환에 있어서의 Prime Spectrum에 관하여 다음 세가지 사실을 증명하였다. 1. X를 환 R의 prime spectrum, C(X)를 X에서 정의되는 실연적함수의 환, X를 C(X)의 maximal spectrum이라 하면 X는 C(X)의 prime spectrum의 부분공간으로서의 한 T-space로 된다. N을 환 R의 nilradical이라 하면, R/N이 regula 이면 X와 X는 위상동형이다. 2. f: R$\longrightarrow$R'을 ring homomorphism, P를 R의 한 Prime ideal, $R_{p}$, R'$_{p}$를 각각 S=R-P 및 f(S)에 관한 분수환(ring of fraction)이라 하고, k(P)를 local ring $R_{p}$의 residue' field라 할 때, R'의 prime spectrum의 부분공간인 $f^{*-1}$(P)는 k(P)(equation omitted)$_{R}$R'의 prime spectrum과 위상동형이다. 단 f*는 f*(Q)=$f^{-1}$(Q)로서 정의되는 함수 s*:Spec(R')$\longrightarrow$Spec(R)이다. 3. X를 환 S의 prime spectrum, N을 R의 nilradical이라 할 때, 다음 네가지 사실은 동치이다. (1) R/N 은 regular 이다. (2) X는 Zarski topology에 관하여 Hausdorff 공간이다. (3) X에서의 Zarski topology와 constructible topology와는 일치한다. (4) R의 임의의 원소 f에 대하여 f를 포함하지 않는 R의 prime ideal 전체의 집합 $X_{f}$는 Zarski topology에 관하여 개집합인 동시에 폐집합이다.폐집합이다....

• Kyungpook Mathematical Journal
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• v.52 no.2
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• pp.149-153
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• 2012

We show that the above lemma and its well-known refinement are valid, in a general setting, in non-commutative rings. Some interesting consequences are also observed.