• Bulletin of the Korean Mathematical Society
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• v.35 no.3
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• pp.455-464
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• 1998

In this paper, we give another proof of Theorem 2.13 of [4] without using the sup property. For the homomorphic image $f(\mu)$ and preimage $f^{-1}(\nu)$ of fuzzy left (resp. right) ideals $\mu$ and $\nu$ respectively, we establish the chains of level left (resp. right) ideals of $f(\mu)$ and $f^{-1}(\nu)$, respectively. Moreover, we prove that a necessary condition for a fuzzy ideal $\mu$ of a near-ring $R$ to be prime is that $\mu$ is two-valued.

• Bulletin of the Korean Mathematical Society
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• v.48 no.6
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• pp.1253-1259
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• 2011

Let R be a prime ring, I a nonzero ideal of R and n a fixed positive integer. If R admits a generalized derivation F associated with a derivation d such that c for all x, $y{\in}I$. Then either R is commutative or n = 1, d = 0 and F is the identity map on R. Moreover in case R is a semiprime ring and $(F([x,\;y]))^n=[x,\;y]$ for all x, $y{\in}R$, then either R is commutative or n = 1, $d(R){\subseteq}Z(R)$, R contains a non-zero central ideal and for all $x{\in}R$.

• Communications of the Korean Mathematical Society
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• v.28 no.1
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• pp.79-85
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• 2013

The structure of a poset P with smallest element 0 is looked at from two view points. Firstly, with respect to the Zariski topology, it is shown that Spec(P), the set of all prime semi-ideals of P, is a compact space and Max(P), the set of all maximal semi-ideals of P, is a compact $T_1$ subspace. Various other topological properties are derived. Secondly, we study the semi-ideal-based zero-divisor graph structure of poset P, denoted by $G_I$ (P), and characterize its diameter.

• Journal of the Chungcheong Mathematical Society
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• v.31 no.1
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• pp.47-52
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• 2018

Let ${\delta}$ be a derivation in a noetherian integral domain A. It is shown that a natural map induces a homeomorphism between the spectrum of $A[z;{\delta}]$ and the Poisson spectrum of $A[z;{\delta}]_p$ such that its restriction to the primitive spectrum of $A[z;{\delta}]$ is also a homeomorphism onto the Poisson primitive spectrum of $A[z;{\delta}]_p$.

• Kyungpook Mathematical Journal
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• v.45 no.4
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• pp.455-470
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• 2005

This paper introduces compactness notions for frames which are expressed in terms of the convergence of suitably specified general filters. It establishes several preservation properties for them as well as their coreflectiveness in the setting of regular frames. Further, it shows that supercompact, compact, and $Lindel{\ddot{o}}f$ frames can be described by compactness conditions of the present form so that various familiar facts become consequences of these general results. In addition, the Prime Ideal Theorem and the Axiom of Countable Choice are proved to be equivalent to certain conditions connected with the kind of compactness considered here.

• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.359-370
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• 2022

The principal aim of this paper is to study the connection between the structure of a quotient ring R/P and the behavior of special additive mappings of R. More precisely, we characterize the commutativity of R/P using derivations (generalized derivations) of R satisfying algebraic identities involving the prime ideal P. Furthermore, we provide examples to show that the various restrictions imposed in the hypothesis of our theorems are not superfluous.

• Journal of applied mathematics & informatics
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• v.42 no.3
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• pp.539-548
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• 2024

Let R be a ring and P be a prime ideal of R. A mapping d : R → R is called a multiplicative derivation if d(xy) = d(x)y + xd(y) for all x, y ∈ R. In this paper, our main motive is to obtain the well-known theorem due to Posner in the ring R/P for generalized derivations associated with a multiplicative derivation defined by an additive mapping F : R → R such that F(xy) = F(x)y + xd(y), where d : R → R is a multiplicative derivation not necessarily additive. This article discusses the use of generalized derivations associated with a multiplicative derivation to investigate the commutativity of the quotient ring R/P.

• Korean Journal of Mathematics
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• v.17 no.1
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• pp.15-23
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• 2009

Let D be an integral domain, P be a nonzero prime ideal of D, $\{P_{\alpha}{\mid}{\alpha}{\in}{\mathcal{A}}\}$ be a nonempty set of prime ideals of D, and $\{I_{\beta}{\mid}{\beta}{\in}{\mathcal{B}}\}$ be a nonempty family of ideals of D with ${\cap}_{{\beta}{\in}{\mathcal{B}}}I_{\beta}{\neq}(0)$. Consider the following conditions: (i) If $P{\subseteq}{\cup}_{{\alpha}{\in}{\mathcal{A}}}P_{\alpha}$, then $P=P_{\alpha}$ for some ${\alpha}{\in}{\mathcal{A}}$; (ii) If ${\cap}_{{\beta}{\in}{\mathcal{B}}}I_{\beta}{\subseteq}P$, then $I_{\beta}{\subseteq}P$ for some ${\beta}{\in}{\mathcal{B}}$. In this paper, we prove that D satisfies $(i){\Leftrightarrow}D$ is a generalized weakly factorial domain of ${\dim}(D)=1{\Rightarrow}D$ satisfies $(ii){\Leftrightarrow}D$ is a weakly Krull domain of dim(D) = 1. We also study the t-operation analogs of (i) and (ii).

• Communications of the Korean Mathematical Society
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• v.35 no.3
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• pp.723-731
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• 2020

The purpose of this paper is to study the structure of quotient rings R/P where R is an arbitrary ring and P is a prime ideal of R. Especially, we will establish a relationship between the structure of this class of rings and the behavior of derivations satisfying algebraic identities involving prime ideals. Furthermore, the characteristic of the quotient ring R/P has been determined in some situations.

• Bulletin of the Korean Mathematical Society
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• v.51 no.1
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• pp.253-266
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• 2014

Commutative rings in which every prime ideal is an intersection of maximal ideals are called Hilbert (or Jacobson) rings. We propose to define classical Hilbert modules by the property that classical prime submodules are intersections of maximal submodules. It is shown that all co-semisimple modules as well as all Artinian modules are classical Hilbert modules. Also, every module over a zero-dimensional ring is classical Hilbert. Results illustrating connections amongst the notions of classical Hilbert module and Hilbert ring are also provided. Rings R over which all modules are classical Hilbert are characterized. Furthermore, we determine the Noetherian rings R for which all finitely generated R-modules are classical Hilbert.