• 충청수학회지
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• 제2권1호
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• pp.37-44
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• 1989

In this paper, it is shown that automatic continuity of derivations on some semi prime Banach algebras can be extended to higher derivations. In particular, we show that if every prime ideal is closed in a commutative semi prime Banach algebra then every higher derivation on it is continuous.

• Kyungpook Mathematical Journal
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• 제55권4호
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• pp.827-835
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• 2015

Let U be a ${\sigma}$-square closed Lie ideal of a 2-torsion free ${\sigma}$-prime ${\Gamma}$-ring M. Let $d{\neq}1$ be an automorphism of M such that $[u,d(u)]_{\alpha}{\in}Z(M)$ on U, $d{\sigma}={\sigma}d$ on U, and there exists $u_0$ in $Sa_{\sigma}(M)$ with $M{\Gamma}u_0{\subseteq}U$. Then, $U{\subseteq}Z(M)$. By applying this result, we generalize the results of Oukhtite and Salhi respect to ${\Gamma}$-rings. Finally, for a non-zero derivation of a 2-torsion free ${\sigma}$-prime $\Gamma$-ring, we obtain suitable conditions under which the $\Gamma$-ring must be commutative.

• Journal of applied mathematics & informatics
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• 제40권3_4호
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• pp.445-460
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• 2022

In this paper, we studied on 𝛽-fuzzy filters of Stone almost distributive lattices. An isomorphism between the lattice of 𝛽-fuzzy filters of a Stone ADL A onto the lattice of fuzzy ideals of the set of all boosters of A is established. The fact that any 𝛽-fuzzy filter of A is an e-fuzzy filter of A is proved. We discuss on some properties of prime 𝛽-fuzzy filters and some topological concepts on the collection of prime 𝛽-fuzzy filters of a Stone ADL. Further we show that the collection 𝓣 = {X^𝛽(λ) : λ is a fuzzy ideal of A} is a topology on 𝓕Spec_𝛽(A) where X^𝛽(λ) = {𝜇 ∈ 𝓕Spec_𝛽(A) : λ ⊈ 𝜇}.

• 대한수학회논문집
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• 제19권2호
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• pp.211-217
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• 2004

In a ring $R_n(K,\;J)$ where K is a commutative ring with identity and J is an ideal of K, all prime ideals of $R_n(K,\;J)$ are of the form either $M_n(P)\;o;R_n(P,\;P\;{\cap}\;J)$. Therefore there is a one to one correspondence between prime ideals of K not containing J and prime ideals of $R_n(K,\;J)$.

• Korean Journal of Mathematics
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• 제8권1호
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• pp.43-45
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• 2000

In this paper, we give the characterization of a left (right) filter of $po$-semigroups.

• 대한수학회논문집
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• 제20권3호
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• pp.427-436
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• 2005

Let (R, m) be a 2-dimensional regular local ring with algebraically closed residue field R/m. Let K be the quotient field of R and v be a prime divisor of R, i.e., a valuation of K which is birationally dominating R and residually transcendental over R. Zariski showed that there are finitely many simple v-ideals $m=P_0\;{\supset}\;P_1\;{\supset}\;{\cdotS}\;{\supset}\;P_t=P$ and all the other v-ideals are uniquely factored into a product of those simple ones. It then was also shown by Lipman that the predecessor of the smallest simple v-ideal P is either simple (P is free) or the product of two simple v-ideals (P is satellite), that the sequence of v-ideals between the maximal ideal and the smallest simple v-ideal P is saturated, and that the v-value of the maximal ideal is the m-adic order of P. Let m = (x, y) and denote the v-value difference |v(x) - v(y)| by $n_v$. In this paper, if the m-adic order of P is 2, we show that $O(P_i)\;=\;1\;for\;1\;{\leq}\;i\; {\leq}\;{\lceil}\;{\frac{b+1}{2}}{\rceil}\;and\;O(P_i)\;=2\;for\;{\lceil}\;\frac{b+3}{2}\rceil\;{\leq}\;i\;\leq\;t,\;where\;b=n_v$. We also show that $n_w\;=\;n_v$ when w is the prime divisor associated to a simple v-ideal $Q\;{\supset}\;P$ of order 2 and that w(R) = v(R) as well.

• 충청수학회지
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• 제26권4호
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• pp.781-786
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• 2013

Let * be a star-operation on an integral domain R. We show that if R is a $*_{\omega}$-Noetherian domain, then quasi-$*_{\omega}$-invertible prime $*_{\omega}$-ideals of R are minimal, and prime ideals of R minimal over a $*_{\omega}$-invertible $*_{\omega}$-ideal are minimal.

• 대한수학회논문집
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• 제15권3호
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• pp.511-519
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• 2000

If G is a strict generalized orthomodular lattice and H={I｜I=[0, $\chi$, $\chi$$\in$G}, then H is prime ideal of the Janowitz's hull J(G) of G. If f is the janowitz's embedding, then the set of all commutatiors of f(G) equals the set of all commutators of the Janowitz's hull J(G) of G. Let L be an OML. Then L J(G) for a strict GOML G if and only if ther exists a proper nonprincipal prime ideal G in L.

• 대한수학회논문집
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• 제16권4호
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• pp.573-583
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• 2001

In this paper we introduce the notion of quotient in-cline and obtain the structure of incline algebra. Moreover, we also introduce the notion of prime and maximal ideal in incline, and study some relations between them in incline algebra.

• 대한수학회보
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• 제56권4호
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• pp.1041-1057
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• 2019

Let ${\Gamma}$ be a nonzero commutative cancellative monoid (written additively), $R={\bigoplus}_{{\alpha}{\in}{\Gamma}}$ $R_{\alpha}$ be a ${\Gamma}$-graded integral domain with $R_{\alpha}{\neq}\{0\}$ for all ${\alpha}{\in}{\Gamma}$, and $S(H)=\{f{\in}R{\mid}C(f)=R\}$. In this paper, we study homogeneously divisorial domains which are graded integral domains whose nonzero homogeneous ideals are divisorial. Among other things, we show that if R is integrally closed, then R is a homogeneously divisorial domain if and only if $R_{S(H)}$ is an h-local $Pr{\ddot{u}}fer$ domain whose maximal ideals are invertible, if and only if R satisfies the following four conditions: (i) R is a graded-$Pr{\ddot{u}}fer$ domain, (ii) every homogeneous maximal ideal of R is invertible, (iii) each nonzero homogeneous prime ideal of R is contained in a unique homogeneous maximal ideal, and (iv) each homogeneous ideal of R has only finitely many minimal prime ideals. We also show that if R is a graded-Noetherian domain, then R is a homogeneously divisorial domain if and only if $R_{S(H)}$ is a divisorial domain of (Krull) dimension one.