• Journal of the Korean Mathematical Society
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• v.42 no.5
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• pp.933-948
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• 2005

Let R be a commutative ring with identity and let M be an R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = 1M. Let M be a non-zero multiplication R-module. Then we prove the following: (1) there exists a bijection: N(M)$\bigcap$V(ann$\_{R}$(M))$\rightarrow$Spec$\_{R}$(M) and in particular, there exists a bijection: N(M)$\bigcap$Max(R)$\rightarrow$Max$\_{R}$(M), (2) N(M) $\bigcap$ V(ann$\_{R}$(M)) = Supp(M) $\bigcap$ V(ann$\_{R}$(M)), and (3) for every ideal I of R, The ideal $\theta$(M) = $\sum$$\_{m(Rm :R M) of R has proved useful in studying multiplication modules. We generalize this ideal to prove the following result: Let R be a commutative ring with identity, P $\in$ Spec(R), and M a non-zero R-module satisfying (1) M is a finitely generated multiplication module, (2) PM is a multiplication module, and (3) P$^{n}$M$\neq$P$^{n+1}$ for every positive integer n, then $\bigcap$$^{$\_{n=1}$(P$^{n}$ + ann$\_{R}$(M)) $\in$ V(ann$\_{R}$(M)) = Supp(M) $\subseteq$ N(M).

• Journal of the Korean Mathematical Society
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• v.44 no.4
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• pp.987-995
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• 2007

We investigate some results which concern the types of Noetherian local rings. In particular, we show that if r(Ap) ${\le}$ depth Ap + 1 for each prime ideal p of a quasi-unmixed Noetherian local ring A, then A is Cohen-Macaulay. It is also shown that the Kawasaki conjecture holds when dim A ${\le}$ depth A + 1. At the end, we deal with some analogous results for modules, which are derived from the results studied on rings.

• Honam Mathematical Journal
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• v.33 no.3
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• pp.419-424
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• 2011

Let D be an integral domain, Spec(D) the set of prime ideals of D, and X a subspace of the Zariski topology on Spec(D). We show that X is compact if and only if given any ideal I of D with $I{\nsubseteq}P$ for all $P{\in}X$, there exists a finitely generated idea $J{\subseteq}I$ such that $J{\nsubseteq}P$ for all $P{\in}X$. We also prove that if D = ${\cap}_{P{\in}X}D_P$ and if * is the star-operation on D induced by X, then X is compact if and only if * $_f$-Max(D) ${\subseteq}$X. As a corollary, we have that t-Max(D) is compact and that ${\mathcal{P}}$(D) = {P${\in}$ Spec(D)$|$P is minimal over (a : b) for some a, b${\in}$D} is compact if and only if t-Max(D) ${\subseteq}\;{\mathcal{P}}$(D).

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.659-668
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• 2021

Let R be a semiprime ring with maximal right ring of quotients Q_mr(R), and let n₁, n₂, …, n_k be k fixed positive integers. Suppose that R is (n₁+n₂+⋯+n_k)!-torsion free, and that f : 𝜌 → Q_mr(R) is an additive map, where 𝜌 is a nonzero right ideal of R. It is proved that if [[…[f(x), x^n₁], …], x^n_k] = 0 for all x ∈ 𝜌, then [f(x), x] = 0 for all x ∈ 𝜌. This gives the result of Beidar et al. [2] for semiprime rings. Moreover, it is also proved that if R is p-torsion, where p is a prime integer with p = Σ^k_i=1 n_i and if f : R → Q_mr(R) is an additive map satisfying [[…[f(x), x^n₁], …], x^n_k] = 0 for all x ∈ R, then [f(x), x] = 0 for all x ∈ R.

• The Journal of Korean Institute of Communications and Information Sciences
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• v.28 no.9C
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• pp.835-842
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• 2003

For an odd prime p and integer n, m and k such that n=(2m＋1)ㆍk, a new family of p-ary sequences of period p$^{n}$ -1 with optimal correlation property is constructed using the p-ary Helleseth-Gong sequences with ideal autocorrelation, where the size of the sequence family is p$^{n}$ . That is, the maximum nontrivial correlation value R$_{max}$ of all pairs of distinct sequences in the family does not exceed p$^{n}$ 2/ ＋1, which means the optimal correlation property in terms of Welch's lower bound. It is also derived that the linear span of the sequences in the family is (m＋2)ㆍn except for the m-sequence in the family.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.311-317
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• 2018

Let (R, m) be a commutative Noetherian local ring and I be an ideal of R. In this paper it is shown that if cd(I, R) = t > 0 and the R-module $Hom_R(R/I,H^t_I(R))$ is finitely generated, then $$t={\sup}\{{\dim}{\widehat{\hat{R}_p}}/Q:p{\in}V(I{\hat{R}}),\;Q{\in}mAss{_{\widehat{\hat{R}_p}}}{\widehat{\hat{R}_p}}\;and\;p{\widehat{\hat{R}_p}}=Rad(I{\wideha{\hat{R}_p}}=Q)\}$$. Moreover, some other results concerning the cohomological dimension of ideals with respect to the rings extension $R{\subset}R[X]$ will be included.

• Kyungpook Mathematical Journal
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• v.51 no.1
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• pp.1-10
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• 2011

Let (R, T) be a normal pair of commutative rings (i.e., R ${\subseteq}$ T is a unita extension of commutative rings, not necessarily integral domains, such that S is integrally closed in T for each ring S such that R ${\subseteq}$ S ${\subseteq}$ T) such that the total quotient ring of R is a von Neumann regular ring. Let P be one of the following ring-theoretic properties: going-down ring, extensionally going-down (EGD) ring, locally divided ring. Then R has P if and only if T has P. An example shows that the "if" part of the assertion fails if P is taken to be the "divided domain" property.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.187-204
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• 2018

Let $R={\bigoplus}_{{\alpha}{\in}{\Gamma}}\;R_{\alpha}$ be a graded integral domain, H be the set of nonzero homogeneous elements of R, and ${\star}$ be a semistar operation on R. The purpose of this paper is to study the properties of $quasi-Pr{\ddot{u}}fer$ and UMt-domains of graded integral domains. For this reason we study the graded analogue of ${\star}-quasi-Pr{\ddot{u}}fer$ domains called $gr-{\star}-quasi-Pr{\ddot{u}}fer$ domains. We study several ring-theoretic properties of $gr-{\star}-quasi-Pr{\ddot{u}}fer$ domains. As an application we give new characterizations of UMt-domains. In particular it is shown that R is a $gr-t-quasi-Pr{\ddot{u}}fer$ domain if and only if R is a UMt-domain if and only if RP is a $quasi-Pr{\ddot{u}}fer$ domain for each homogeneous maximal t-ideal P of R. We also show that R is a UMt-domain if and only if H is a t-splitting set in R[X] if and only if each prime t-ideal Q in R[X] such that $Q{\cap}H ={\emptyset}$ is a maximal t-ideal.

• Cross-Cultural Studies
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• v.24
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• pp.19-31
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• 2011

This article examines the historical view of Benito $P{\acute{e}}rez$ $Gald{\acute{o}}s$ in his historical novel named "Zaragoza" and "$C{\acute{a}}novas$" by comparing with Numancia of Cervantes. In front of the national survival crisis, $Gald{\acute{o}}s$ recognizes that it is necessary that Spain should recover humanity and morality on basis of the krausist ideal. The krausists warned subjective idealism, stressing on free-will and moral conscience in a harmonious balance of rationality. Without refusing patriotism, they warned egoistic patriotism deformed by national selfishness or nationalism. Giner de los Ríos insisted on a necessity of harmonizing national inclination and universal one. $Gald{\acute{o}}s$ emphasizes that the identity of Spain is not possession of privileged class but sweat and tears of the mass of people in daily life. He represents Efemera as the ideal history on the basis of Unamuno's 'intrahistoria', by which, instead of $Luk{\acute{a}}cs^{\prime}$ progressive history, we can peep into a historical poetics of human free nature and vividness of discontinuous and endless shifting. According to the sub-history, we come to the conclusion that $Gald{\acute{o}}s$ and Cervantes would lay emphasis on sub-history, aiming to essence of Spanish soul originated from national landscape and people's daily life.

• Korean Journal of Mathematics
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• v.27 no.2
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• pp.375-415
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• 2019

We study spaces of continuous-function-valued functions that have the property that composition with evaluation functionals induce $weak^*$ to norm continuous maps to ${\ell}^p$ space ($p{\in}(1,\;{\infty})$). Versions of $H{\ddot{o}}lder^{\prime}s$ inequality and Riesz representation theorem are proved to hold on these spaces. We prove a version of Dixmier's theorem for spaces of function-valued matrix operators on these spaces, and an analogue of the trace formula for operators on Hilbert spaces. When the function space is taken to be the complex field, the spaces are just the ${\ell}^p$ spaces and the well-known classical theorems follow from our results.